{
  "index": "1952-B-7",
  "type": "ANA",
  "tag": [
    "ANA",
    "ALG"
  ],
  "difficulty": "",
  "question": "7. Given any real number \\( N_{0} \\), if \\( N_{j+1}=\\cos N_{j} \\), prove that \\( \\lim _{j-\\infty} N_{j} \\) exists and is independent of \\( N_{0} \\).",
  "solution": "Solution. The graphical method for recursions (explained on p. 223) makes it clear that for any choice of \\( N_{0} \\), the sequence converges to the unique root of the equation \\( x=\\cos x \\). We formalize this analysis.\n\nSince \\( f(x)=x-\\cos x \\) defines a strictly increasing function (its derivative is non-negative with only isolated zeros), \\( f(0)<0 \\), and \\( f(1)>0 \\), there is a unique number \\( \\xi \\) in \\( (0,1) \\) such that \\( f(\\xi)=0 \\); i.e., \\( \\cos \\xi=\\xi \\).\n\nBy the mean value theorem, for any \\( x \\) there is an \\( \\eta \\) between \\( x \\) and \\( \\xi \\) such that\n\\[\n\\cos x-\\cos \\xi=-(\\sin \\eta)(x-\\xi)\n\\]\n\nIf \\( x \\in[0,1] \\), then \\( \\eta \\in(0,1) \\) and \\( |\\sin \\eta|<\\sin 1 \\); hence,\n\\[\n|\\cos x-\\cos \\xi| \\leq(\\sin 1)|x-\\xi|\n\\]\n\nNow \\( N_{1}=\\cos N_{0} \\in[-1,1], N_{2}=\\cos N_{1} \\in[0,1] \\), and \\( N_{j} \\in[0,1] \\) for \\( j \\geq 2 \\) by induction. Hence (1) yields\n\\[\n\\left|N_{j+1}-\\xi\\right|=\\left|\\cos N_{j}-\\cos \\xi\\right| \\leq(\\sin 1)\\left|N_{j}-\\xi\\right|\n\\]\nfor \\( j \\geq 2 \\), and therefore\n\\[\n\\left|N_{j}-\\xi\\right| \\leq(\\sin 1)^{i}{ }^{2}\\left|N_{2}-\\xi\\right|\n\\]\nfor all \\( j \\geq 2 \\). Since \\( \\sin 1<1 \\), it follows that \\( \\left|N_{j}-\\xi\\right| \\rightarrow 0 \\) and therefore\n\\[\n\\lim _{j \\rightarrow \\infty} N_{j}=\\xi .\n\\]\n\nRemarks. Using the recursion and a hand calculator, it is easy to find \\( \\boldsymbol{\\xi} \\) to as many decimals as the capacity of the machine. In fact, \\( \\boldsymbol{\\xi} \\sim 0.739085 \\).\n\nThe problem is an example of a very general theorem about recursions. Suppose \\( f \\) is any function of class \\( C^{\\prime}, f(\\xi)=\\xi \\), and \\( \\left|f^{\\prime}(\\xi)\\right|<1 \\). Then the recursion\n\\[\nx_{n+1}=f\\left(x_{n}\\right)\n\\]\nproduces a sequence \\( \\left\\{x_{n}\\right\\} \\) that converges to \\( \\xi \\), provided \\( x_{0} \\), or some later member of the sequence, is sufficiently close to \\( \\xi \\). Precisely, if \\( I \\) is an interval symmetric about \\( \\xi \\) on which \\( \\left|f^{\\prime}\\right| \\leq \\alpha<1 \\), and \\( x_{k} \\in I \\), then\n\\[\n\\left|x_{n}-\\xi\\right| \\leq \\alpha^{n}{ }^{k}\\left|x_{k}-\\xi\\right|,\n\\]\nfor \\( n>k \\), so \\( x_{n} \\rightarrow \\xi \\). In this situation, \\( \\xi \\) is said to be an attractive fixed point of \\( f \\).\n\nThere is a vast literature on the subject of calculating roots of equations by iteration. See, for example, Ortega and Rheinboldt, Iterative Solutions of Non-linear Equations in Several Variables, Academic Press, New York, 1970, pages \\( 120-125 \\).",
  "vars": [
    "N_0",
    "N_j+1",
    "N_j",
    "x",
    "j",
    "N_1",
    "N_2",
    "x_n+1",
    "x_n",
    "x_k",
    "n",
    "k",
    "\\\\eta"
  ],
  "params": [
    "\\\\xi",
    "f",
    "I",
    "\\\\alpha"
  ],
  "sci_consts": [],
  "variants": {
    "descriptive_long": {
      "map": {
        "N_0": "initval",
        "N_j+1": "nextrec",
        "N_j": "currentrec",
        "x": "genericx",
        "j": "iterindex",
        "N_1": "firstrec",
        "N_2": "secondrec",
        "x_n+1": "seqplusone",
        "x_n": "seqvalue",
        "x_k": "subseqterm",
        "n": "mainindex",
        "k": "subindexk",
        "\\eta": "meanvar",
        "\\xi": "fixpoint",
        "f": "functionf",
        "I": "intervali",
        "\\alpha": "alphabound"
      },
      "question": "7. Given any real number \\( initval \\), if \\( nextrec=\\cos currentrec \\), prove that \\( \\lim _{iterindex-\\infty} currentrec \\) exists and is independent of \\( initval \\).",
      "solution": "Solution. The graphical method for recursions (explained on p. 223) makes it clear that for any choice of \\( initval \\), the sequence converges to the unique root of the equation \\( genericx=\\cos genericx \\). We formalize this analysis.\n\nSince \\( functionf(genericx)=genericx-\\cos genericx \\) defines a strictly increasing function (its derivative is non-negative with only isolated zeros), \\( functionf(0)<0 \\), and \\( functionf(1)>0 \\), there is a unique number \\( fixpoint \\) in \\( (0,1) \\) such that \\( functionf(fixpoint)=0 \\); i.e., \\( \\cos fixpoint=fixpoint \\).\n\nBy the mean value theorem, for any \\( genericx \\) there is a \\( meanvar \\) between \\( genericx \\) and \\( fixpoint \\) such that\n\\[\n\\cos genericx-\\cos fixpoint=-(\\sin meanvar)(genericx-fixpoint)\n\\]\nIf \\( genericx \\in[0,1] \\), then \\( meanvar \\in(0,1) \\) and \\( |\\sin meanvar|<\\sin 1 \\); hence,\n\\[\n|\\cos genericx-\\cos fixpoint| \\leq(\\sin 1)|genericx-fixpoint|\n\\]\nNow \\( firstrec=\\cos initval \\in[-1,1],\\; secondrec=\\cos firstrec \\in[0,1] \\), and \\( currentrec \\in[0,1] \\) for \\( iterindex \\geq 2 \\) by induction. Hence (1) yields\n\\[\n\\left|nextrec-fixpoint\\right|=\\left|\\cos currentrec-\\cos fixpoint\\right| \\leq(\\sin 1)\\left|currentrec-fixpoint\\right|\n\\]\nfor \\( iterindex \\geq 2 \\), and therefore\n\\[\n\\left|currentrec-fixpoint\\right| \\leq(\\sin 1)^{i}{ }^{2}\\left|secondrec-fixpoint\\right|\n\\]\nfor all \\( iterindex \\geq 2 \\). Since \\( \\sin 1<1 \\), it follows that \\( \\left|currentrec-fixpoint\\right| \\rightarrow 0 \\) and therefore\n\\[\n\\lim _{iterindex \\rightarrow \\infty} currentrec=fixpoint .\n\\]\n\nRemarks. Using the recursion and a hand calculator, it is easy to find \\( \\boldsymbol{fixpoint} \\) to as many decimals as the capacity of the machine. In fact, \\( \\boldsymbol{fixpoint} \\sim 0.739085 \\).\n\nThe problem is an example of a very general theorem about recursions. Suppose \\( functionf \\) is any function of class \\( C^{\\prime} ,\\; functionf(fixpoint)=fixpoint \\), and \\( \\left|functionf^{\\prime}(fixpoint)\\right|<1 \\). Then the recursion\n\\[\nseqplusone=functionf\\left(seqvalue\\right)\n\\]\nproduces a sequence \\( \\{seqvalue\\} \\) that converges to \\( fixpoint \\), provided \\( genericx_{0} \\), or some later member of the sequence, is sufficiently close to \\( fixpoint \\). Precisely, if \\( intervali \\) is an interval symmetric about \\( fixpoint \\) on which \\( \\left|functionf^{\\prime}\\right| \\leq alphabound<1 \\), and \\( subseqterm \\in intervali \\), then\n\\[\n\\left|seqvalue-fixpoint\\right| \\leq alphabound^{mainindex}{ }^{subindexk}\\left|subseqterm-fixpoint\\right|,\n\\]\nfor \\( mainindex>subindexk \\), so \\( seqvalue \\rightarrow fixpoint \\). In this situation, \\( fixpoint \\) is said to be an attractive fixed point of \\( functionf \\).\n\nThere is a vast literature on the subject of calculating roots of equations by iteration. See, for example, Ortega and Rheinboldt, Iterative Solutions of Non-linear Equations in Several Variables, Academic Press, New York, 1970, pages \\( 120-125 \\)."
    },
    "descriptive_long_confusing": {
      "map": {
        "N_0": "compassion",
        "N_j+1": "pineapple",
        "N_j": "shoelaces",
        "x": "butterfly",
        "j": "rainstorm",
        "N_1": "strawberry",
        "N_2": "chocolate",
        "x_n+1": "skylights",
        "x_n": "porcupine",
        "x_k": "blueberry",
        "n": "sidewalk",
        "k": "backpack",
        "\\eta": "daggering",
        "\\xi": "watershed",
        "f": "lighthouse",
        "I": "cellarway",
        "\\alpha": "moonlight"
      },
      "question": "7. Given any real number \\( compassion \\), if \\( pineapple=\\cos shoelaces \\), prove that \\( \\lim _{rainstorm-\\infty} shoelaces \\) exists and is independent of \\( compassion \\).",
      "solution": "Solution. The graphical method for recursions (explained on p. 223) makes it clear that for any choice of \\( compassion \\), the sequence converges to the unique root of the equation \\( butterfly=\\cos butterfly \\). We formalize this analysis.\n\nSince \\( lighthouse(butterfly)=butterfly-\\cos butterfly \\) defines a strictly increasing function (its derivative is non-negative with only isolated zeros), \\( lighthouse(0)<0 \\), and \\( lighthouse(1)>0 \\), there is a unique number \\( watershed \\) in \\( (0,1) \\) such that \\( lighthouse(watershed)=0 \\); i.e., \\( \\cos watershed=watershed \\).\n\nBy the mean value theorem, for any \\( butterfly \\) there is an \\( daggering \\) between \\( butterfly \\) and \\( watershed \\) such that\n\\[\n\\cos butterfly-\\cos watershed=-(\\sin daggering)(butterfly-watershed)\n\\]\n\nIf \\( butterfly \\in[0,1] \\), then \\( daggering \\in(0,1) \\) and \\( |\\sin daggering|<\\sin 1 \\); hence,\n\\[\n|\\cos butterfly-\\cos watershed| \\leq(\\sin 1)|butterfly-watershed|\n\\]\n\nNow \\( strawberry=\\cos compassion \\in[-1,1],\\; chocolate=\\cos strawberry \\in[0,1] \\), and \\( shoelaces \\in[0,1] \\) for \\( rainstorm \\geq 2 \\) by induction. Hence (1) yields\n\\[\n\\left|pineapple-watershed\\right|=\\left|\\cos shoelaces-\\cos watershed\\right| \\leq(\\sin 1)\\left|shoelaces-watershed\\right|\n\\]\nfor \\( rainstorm \\geq 2 \\), and therefore\n\\[\n\\left|shoelaces-watershed\\right| \\leq(\\sin 1)^{i}{ }^{2}\\left|chocolate-watershed\\right|\n\\]\nfor all \\( rainstorm \\geq 2 \\). Since \\( \\sin 1<1 \\), it follows that \\( \\left|shoelaces-watershed\\right| \\rightarrow 0 \\) and therefore\n\\[\n\\lim _{rainstorm \\rightarrow \\infty} shoelaces=watershed .\n\\]\n\nRemarks. Using the recursion and a hand calculator, it is easy to find \\( \\boldsymbol{watershed} \\) to as many decimals as the capacity of the machine. In fact, \\( \\boldsymbol{watershed} \\sim 0.739085 \\).\n\nThe problem is an example of a very general theorem about recursions. Suppose \\( lighthouse \\) is any function of class \\( C^{\\prime}, lighthouse(watershed)=watershed \\), and \\( \\left|lighthouse^{\\prime}(watershed)\\right|<1 \\). Then the recursion\n\\[\nskylights=lighthouse(porcupine)\\]\nproduces a sequence \\( \\left\\{porcupine\\right\\} \\) that converges to \\( watershed \\), provided \\( x_{0} \\), or some later member of the sequence, is sufficiently close to \\( watershed \\). Precisely, if \\( cellarway \\) is an interval symmetric about \\( watershed \\) on which \\( \\left|lighthouse^{\\prime}\\right| \\leq moonlight<1 \\), and \\( blueberry \\in cellarway \\), then\n\\[\n\\left|porcupine-watershed\\right| \\leq moonlight^{sidewalk}{ }^{backpack}\\left|blueberry-watershed\\right|,\n\\]\nfor \\( sidewalk>backpack \\), so \\( porcupine \\rightarrow watershed \\). In this situation, \\( watershed \\) is said to be an attractive fixed point of \\( lighthouse \\).\n\nThere is a vast literature on the subject of calculating roots of equations by iteration. See, for example, Ortega and Rheinboldt, Iterative Solutions of Non-linear Equations in Several Variables, Academic Press, New York, 1970, pages \\( 120-125 \\)."
    },
    "descriptive_long_misleading": {
      "map": {
        "N_0": "terminalpoint",
        "N_j+1": "previousincrement",
        "N_j": "constantvalue",
        "x": "yaxisvalue",
        "j": "stationary",
        "N_1": "ultimateone",
        "N_2": "ultimatepair",
        "x_n+1": "earlierstep",
        "x_n": "futurestep",
        "x_k": "fixedstage",
        "n": "microindex",
        "k": "rigidindex",
        "\\\\eta": "endpoint",
        "\\\\xi": "divergence",
        "f": "steadfast",
        "I": "singlepoint",
        "\\\\alpha": "omegalast"
      },
      "question": "7. Given any real number \\( terminalpoint \\), if \\( previousincrement=\\cos constantvalue \\), prove that \\( \\lim _{stationary-\\infty} constantvalue \\) exists and is independent of \\( terminalpoint \\).",
      "solution": "Solution. The graphical method for recursions (explained on p. 223) makes it clear that for any choice of \\( terminalpoint \\), the sequence converges to the unique root of the equation \\( yaxisvalue=\\cos yaxisvalue \\). We formalize this analysis.\n\nSince \\( steadfast(yaxisvalue)=yaxisvalue-\\cos yaxisvalue \\) defines a strictly increasing function (its derivative is non-negative with only isolated zeros), \\( steadfast(0)<0 \\), and \\( steadfast(1)>0 \\), there is a unique number \\( divergence \\) in \\( (0,1) \\) such that \\( steadfast(divergence)=0 \\); i.e., \\( \\cos divergence=divergence \\).\n\nBy the mean value theorem, for any \\( yaxisvalue \\) there is an \\( endpoint \\) between \\( yaxisvalue \\) and \\( divergence \\) such that\n\\[\n\\cos yaxisvalue-\\cos divergence=-(\\sin endpoint)(yaxisvalue-divergence)\n\\]\n\nIf \\( yaxisvalue \\in[0,1] \\), then \\( endpoint \\in(0,1) \\) and \\( |\\sin endpoint|<\\sin 1 \\); hence,\n\\[\n|\\cos yaxisvalue-\\cos divergence| \\leq(\\sin 1)|yaxisvalue-divergence|\n\\]\n\nNow \\( ultimateone=\\cos terminalpoint \\in[-1,1], ultimatepair=\\cos ultimateone \\in[0,1] \\), and \\( constantvalue \\in[0,1] \\) for \\( stationary \\geq 2 \\) by induction. Hence (1) yields\n\\[\n\\left|previousincrement-divergence\\right|=\\left|\\cos constantvalue-\\cos divergence\\right| \\leq(\\sin 1)\\left|constantvalue-divergence\\right|\n\\]\nfor \\( stationary \\geq 2 \\), and therefore\n\\[\n\\left|constantvalue-divergence\\right| \\leq(\\sin 1)^{i}{ }^{2}\\left|ultimatepair-divergence\\right|\n\\]\nfor all \\( stationary \\geq 2 \\). Since \\( \\sin 1<1 \\), it follows that \\( \\left|constantvalue-divergence\\right| \\rightarrow 0 \\) and therefore\n\\[\n\\lim _{stationary \\rightarrow \\infty} constantvalue=divergence .\n\\]\n\nRemarks. Using the recursion and a hand calculator, it is easy to find \\( \\boldsymbol{divergence} \\) to as many decimals as the capacity of the machine. In fact, \\( \\boldsymbol{divergence} \\sim 0.739085 \\).\n\nThe problem is an example of a very general theorem about recursions. Suppose \\( steadfast \\) is any function of class \\( C^{\\prime}, steadfast(divergence)=divergence \\), and \\( \\left|steadfast^{\\prime}(divergence)\\right|<1 \\). Then the recursion\n\\[\nearlierstep=steadfast\\left(futurestep\\right)\n\\]\nproduces a sequence \\( \\left\\{futurestep\\right\\} \\) that converges to \\( divergence \\), provided \\( yaxisvalue_{0} \\), or some later member of the sequence, is sufficiently close to \\( divergence \\). Precisely, if \\( singlepoint \\) is an interval symmetric about \\( divergence \\) on which \\( \\left|steadfast^{\\prime}\\right| \\leq omegalast<1 \\), and \\( fixedstage \\in singlepoint \\), then\n\\[\n\\left|futurestep-divergence\\right| \\leq omegalast^{microindex}{ }^{rigidindex}\\left|fixedstage-divergence\\right|,\n\\]\nfor \\( microindex>rigidindex \\), so \\( futurestep \\rightarrow divergence \\). In this situation, \\( divergence \\) is said to be an attractive fixed point of \\( steadfast \\).\n\nThere is a vast literature on the subject of calculating roots of equations by iteration. See, for example, Ortega and Rheinboldt, Iterative Solutions of Non-linear Equations in Several Variables, Academic Press, New York, 1970, pages \\( 120-125 \\)."
    },
    "garbled_string": {
      "map": {
        "N_0": "rkjvbnmns",
        "N_j+1": "qpwmsldke",
        "N_j": "zhqtpbnae",
        "x": "plmareovk",
        "j": "cxzvbnmtyu",
        "N_1": "ghrslkzop",
        "N_2": "vbfjskalm",
        "x_n+1": "hdkslqopr",
        "x_n": "wepqmbnsa",
        "x_k": "jlkasmdqe",
        "n": "reopqlskt",
        "k": "zmxncbvas",
        "\\eta": "hqowpzmnr",
        "\\xi": "lksjdfwqz",
        "f": "mqpwordkj",
        "I": "vcnmxyuio",
        "\\alpha": "qwertyuiop"
      },
      "question": "Given any real number \\( rkjvbnmns \\), if \\( qpwmsldke=\\cos zhqtpbnae \\), prove that \\( \\lim _{cxzvbnmtyu-\\infty} zhqtpbnae \\) exists and is independent of \\( rkjvbnmns \\).",
      "solution": "The graphical method for recursions (explained on p. 223) makes it clear that for any choice of \\( rkjvbnmns \\), the sequence converges to the unique root of the equation \\( plmareovk=\\cos plmareovk \\). We formalize this analysis.\n\nSince \\( mqpwordkj(plmareovk)=plmareovk-\\cos plmareovk \\) defines a strictly increasing function (its derivative is non-negative with only isolated zeros), \\( mqpwordkj(0)<0 \\), and \\( mqpwordkj(1)>0 \\), there is a unique number \\( lksjdfwqz \\) in \\( (0,1) \\) such that \\( mqpwordkj(lksjdfwqz)=0 \\); i.e., \\( \\cos lksjdfwqz=lksjdfwqz \\).\n\nBy the mean value theorem, for any \\( plmareovk \\) there is an \\( hqowpzmnr \\) between \\( plmareovk \\) and \\( lksjdfwqz \\) such that\n\\[\n\\cos plmareovk-\\cos lksjdfwqz=-(\\sin hqowpzmnr)(plmareovk-lksjdfwqz)\n\\]\nIf \\( plmareovk \\in[0,1] \\), then \\( hqowpzmnr \\in(0,1) \\) and \\( |\\sin hqowpzmnr|<\\sin 1 \\); hence,\n\\[\n|\\cos plmareovk-\\cos lksjdfwqz| \\le(\\sin 1)|plmareovk-lksjdfwqz|\n\\]\nNow \\( ghrslkzop=\\cos rkjvbnmns \\in[-1,1], vbfjskalm=\\cos ghrslkzop \\in[0,1] \\), and \\( zhqtpbnae \\in[0,1] \\) for \\( cxzvbnmtyu \\ge 2 \\) by induction. Hence (1) yields\n\\[\n\\left|qpwmsldke-lksjdfwqz\\right|=\\left|\\cos zhqtpbnae-\\cos lksjdfwqz\\right| \\le(\\sin 1)\\left|zhqtpbnae-lksjdfwqz\\right|\n\\]\nfor \\( cxzvbnmtyu \\ge 2 \\), and therefore\n\\[\n\\left|zhqtpbnae-lksjdfwqz\\right| \\le(\\sin 1)^{i}{ }^{2}\\left|vbfjskalm-lksjdfwqz\\right|\n\\]\nfor all \\( cxzvbnmtyu \\ge 2 \\). Since \\( \\sin 1<1 \\), it follows that \\( \\left|zhqtpbnae-lksjdfwqz\\right| \\rightarrow 0 \\) and therefore\n\\[\n\\lim _{cxzvbnmtyu \\rightarrow \\infty} zhqtpbnae=lksjdfwqz .\n\\]\n\nRemarks. Using the recursion and a hand calculator, it is easy to find \\( \\boldsymbol{lksjdfwqz} \\) to as many decimals as the capacity of the machine. In fact, \\( \\boldsymbol{lksjdfwqz} \\sim 0.739085 \\).\n\nThe problem is an example of a very general theorem about recursions. Suppose \\( mqpwordkj \\) is any function of class \\( C^{\\prime}, mqpwordkj(lksjdfwqz)=lksjdfwqz \\), and \\( \\left|mqpwordkj^{\\prime}(lksjdfwqz)\\right|<1 \\). Then the recursion\n\\[\nhdkslqopr=mqpwordkj\\left(wepqmbnsa\\right)\n\\]\nproduces a sequence \\( \\left\\{wepqmbnsa\\right\\} \\) that converges to \\( lksjdfwqz \\), provided \\( x_{0} \\), or some later member of the sequence, is sufficiently close to \\( lksjdfwqz \\). Precisely, if \\( vcnmxyuio \\) is an interval symmetric about \\( lksjdfwqz \\) on which \\( \\left|mqpwordkj^{\\prime}\\right| \\le qwertyuiop<1 \\), and \\( jlkasmdqe \\in vcnmxyuio \\), then\n\\[\n\\left|wepqmbnsa-lksjdfwqz\\right| \\le qwertyuiop^{reopqlskt}{ }^{zmxncbvas}\\left|jlkasmdqe-lksjdfwqz\\right|,\n\\]\nfor \\( reopqlskt>zmxncbvas \\), so \\( wepqmbnsa \\rightarrow lksjdfwqz \\). In this situation, \\( lksjdfwqz \\) is said to be an attractive fixed point of \\( mqpwordkj \\).\n\nThere is a vast literature on the subject of calculating roots of equations by iteration. See, for example, Ortega and Rheinboldt, Iterative Solutions of Non-linear Equations in Several Variables, Academic Press, New York, 1970, pages \\( 120-125 \\)."
    },
    "kernel_variant": {
      "question": "Fix an arbitrary vector\n\\[\nP_{0}=(x_{0},y_{0},z_{0})\\in\\mathbb R^{3}\n\\]\nand define the sequence $\\,(P_{j})_{j\\ge0}$ by the simultaneous recurrences\n\\[\n\\begin{aligned}\nx_{j+1}&=\\frac{2+\\sin x_{j}+\\cos y_{j}+\\tanh z_{j}}{4},\\\\[2pt]\ny_{j+1}&=\\frac{2+\\sin y_{j}+\\cos z_{j}+\\tanh x_{j}}{4},\\\\[2pt]\nz_{j+1}&=\\frac{2+\\sin z_{j}+\\cos x_{j}+\\tanh y_{j}}{4}\\qquad(j=0,1,2,\\dots).\n\\end{aligned}\n\\]\n\n(a)  Prove that the limit $\\displaystyle\\lim_{j\\to\\infty}P_{j}$ exists, is independent of the initial vector $P_{0}$ and has the form $(L,L,L)$.\n\n(b)  Show that the common limit $L$ is the unique real root of the scalar equation\n\\[\n\\boxed{\\;t=\\frac{2+\\sin t+\\cos t+\\tanh t}{4}\\;}\n\\]\nand determine it to four‐decimal accuracy.",
      "solution": "Step 1.  A forward–invariant compact box.  \nBecause $-1\\le\\sin u,\\tanh u\\le1$ and $-1\\le\\cos u\\le1$ for every $u\\in\\mathbb R$,\n\\[\n-1\\;\\le\\;2+\\sin u+\\cos v+\\tanh w\\;\\le\\;5.\n\\]\nDividing by $4$ yields\n\\[\n-0.25\\le x_{1},y_{1},z_{1}\\le1.25 .\n\\tag{1.1}\n\\]\n\nWe claim that the interval $I:=[-\\,0.25,1.25]$ is forward–invariant.  \nAssume $x,y,z\\in I$.  Since $|x|\\le1.25$, we have\n\\[\n-1\\le\\sin x,\\tanh x\\le1,\\qquad\n\\cos x\\ge \\cos1.25\\approx0.315.\n\\]\nHence\n\\[\n0.315=2-1-1+\\cos1.25\\le\n2+\\sin x+\\cos y+\\tanh z\\le5,\n\\]\nand dividing by $4$ gives\n\\[\n0.07875\\le\\frac{2+\\sin x+\\cos y+\\tanh z}{4}\\le1.25 .\n\\]\nThus $F(I^{3})\\subseteq I^{3}$.  Together with (1.1) we conclude\n\\[\n-0.25\\le x_{j},y_{j},z_{j}\\le1.25 \\quad(j\\ge1).\n\\tag{1.2}\n\\]\n(This step is not essential for the contraction argument below, but it provides a convenient global bound.)\n\nStep 2.  Global contraction in the $\\|\\cdot\\|_{\\infty}$–norm.  \nLet $F:\\mathbb R^{3}\\to\\mathbb R^{3}$ denote the right–hand side of the recursion.  Its Jacobian is\n\\[\nDF(P)=\\frac14\n\\begin{pmatrix}\n\\cos x & -\\sin y & \\operatorname{sech}^{2} z\\\\\n\\operatorname{sech}^{2} x & \\cos y & -\\sin z\\\\\n-\\sin x & \\operatorname{sech}^{2} y & \\cos z\n\\end{pmatrix},\n\\qquad P=(x,y,z).\n\\]\nSince $|\\cos|\\le1$, $|\\sin|\\le1$ and $0<\\operatorname{sech}^{2}\\le1$, every entry of $DF(P)$ has absolute value $\\le\\frac14$.  Therefore the maximum row–sum norm satisfies\n\\[\n\\|DF(P)\\|_{\\infty}\\le\\frac14+\\frac14+\\frac14=\\frac34<1\n\\qquad(\\forall P\\in\\mathbb R^{3}),\n\\]\nso $F$ is a global contraction on $(\\mathbb R^{3},\\|\\cdot\\|_{\\infty})$ with Lipschitz constant $q=\\tfrac34$.\n\nStep 3.  Existence and uniqueness of a fixed point.  \nBy the Banach Fixed Point Theorem the contraction $F$ possesses a unique fixed point $P^{*}\\in\\mathbb R^{3}$, and every orbit satisfies\n\\[\n\\|P_{j}-P^{*}\\|_{\\infty}\\le q^{\\,j}\\,\\|P_{0}-P^{*}\\|_{\\infty}\\qquad(j\\ge0).\n\\tag{3.1}\n\\]\n\nStep 4.  Symmetry of the fixed point.  \nBecause $F$ is invariant under the cyclic permutation $(x,y,z)\\mapsto(y,z,x)$, uniqueness of the fixed point implies its invariance, hence\n\\[\nP^{*}=(L,L,L)\\quad\\text{for some }L\\in\\mathbb R.\n\\]\n\nStep 5.  Scalar characterisation of $L$.  \nSubstituting $x=y=z=L$ into the defining recurrence yields exactly\n\\[\nL=\\frac{2+\\sin L+\\cos L+\\tanh L}{4},\n\\]\nwhich is the equation stated in part (b).\n\nStep 6.  Uniqueness of the scalar root.  \nDefine $g(t)=\\frac{2+\\sin t+\\cos t+\\tanh t}{4}$.  Then\n\\[\n|g'(t)|=\\Bigl|\\tfrac14\\bigl(\\cos t-\\sin t+\\operatorname{sech}^{2}t\\bigr)\\Bigr|\n\\le\\tfrac14(1+1+1)=\\frac34<1\n\\qquad(\\forall t\\in\\mathbb R),\n\\]\nso $g$ is a contraction on $\\mathbb R$ with the same constant $q=\\tfrac34$.  Therefore it has a unique fixed point, namely $L$.\n\nStep 7.  Numerical evaluation to four decimals.  \nFor contractions the estimate\n\\[\n|L-t_{k}|\\le\\frac{1}{1-q}\\,|t_{k}-t_{k-1}|\n\\tag{7.1}\n\\]\nholds for the Picard iterates $t_{k+1}=g(t_{k})$.  Because $q=\\tfrac34$, the prefactor is\n\\[\n\\frac1{1-q}=\\frac1{1-\\frac34}=4.\n\\]\n\nStart with $t_{0}=1$ and iterate:\n\n\\[\n\\begin{array}{lcl}\nt_{1}=g(1)&=&1.0358418618,\\\\\nt_{2}=g(t_{1})&=&1.0366500807,\\\\\nt_{3}=g(t_{2})&=&1.0366274868.\n\\end{array}\n\\]\n\nFrom the last two iterates we have $|t_{3}-t_{2}|<2.3\\times10^{-5}$.  Inequality (7.1) therefore gives\n\\[\n|L-t_{3}|<4\\cdot2.3\\times10^{-5}=9.2\\times10^{-5}<10^{-4}.\n\\]\nThus $t_{3}$ is already accurate to four decimal places.  Rounding,\n\\[\n\\boxed{L\\approx1.0366}.\n\\]\nConsequently the sequence $(P_{j})$ issued from any starting point converges to $(1.0366,1.0366,1.0366)$, and the requested four–decimal value is $L=1.0366\\pm0.0001$.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T19:09:31.451603",
        "was_fixed": false,
        "difficulty_analysis": "• Dimensional escalation: the problem moves from a single real recursion to a coupled 3-dimensional nonlinear system; interactions among coordinates must be controlled simultaneously.  \n• Advanced tools: the solution demands Jacobian‐norm estimates, Banach’s Fixed Point Theorem in a normed vector space, and symmetry arguments to force diagonal convergence.  \n• Global (not local) analysis: unlike the original, which only needs an invariant interval, here a uniform Lipschitz bound on all of $\\mathbb R^{3}$ must be established.  \n• Uniqueness reasoning is subtler: cyclic symmetry plus uniqueness of the fixed point is used to deduce its diagonal form.  \n• Additional functions: the presence of $\\tanh$ alongside $\\sin$ and $\\cos$ requires combining trigonometric and hyperbolic estimates.  \nTogether these elements introduce multiple interacting concepts and a deeper theoretical framework, making the enhanced variant significantly harder than both the original problem and the easier kernel variant."
      }
    },
    "original_kernel_variant": {
      "question": "Fix an arbitrary vector\n\\[\nP_{0}=(x_{0},y_{0},z_{0})\\in\\mathbb R^{3}\n\\]\nand define the sequence $\\,(P_{j})_{j\\ge0}$ by the simultaneous recurrences\n\\[\n\\begin{aligned}\nx_{j+1}&=\\frac{2+\\sin x_{j}+\\cos y_{j}+\\tanh z_{j}}{4},\\\\[2pt]\ny_{j+1}&=\\frac{2+\\sin y_{j}+\\cos z_{j}+\\tanh x_{j}}{4},\\\\[2pt]\nz_{j+1}&=\\frac{2+\\sin z_{j}+\\cos x_{j}+\\tanh y_{j}}{4}\\qquad(j=0,1,2,\\dots).\n\\end{aligned}\n\\]\n\n(a)  Prove that the limit $\\displaystyle\\lim_{j\\to\\infty}P_{j}$ exists, is independent of the initial vector $P_{0}$ and has the form $(L,L,L)$.\n\n(b)  Show that the common limit $L$ is the unique real root of the scalar equation\n\\[\n\\boxed{\\;t=\\frac{2+\\sin t+\\cos t+\\tanh t}{4}\\;}\n\\]\nand determine it to four‐decimal accuracy.",
      "solution": "Step 1.  A forward–invariant compact box.  \nBecause $-1\\le\\sin u,\\tanh u\\le1$ and $-1\\le\\cos u\\le1$ for every $u\\in\\mathbb R$,\n\\[\n-1\\;\\le\\;2+\\sin u+\\cos v+\\tanh w\\;\\le\\;5.\n\\]\nDividing by $4$ yields\n\\[\n-0.25\\le x_{1},y_{1},z_{1}\\le1.25 .\n\\tag{1.1}\n\\]\n\nWe claim that the interval $I:=[-\\,0.25,1.25]$ is forward–invariant.  \nAssume $x,y,z\\in I$.  Since $|x|\\le1.25$, we have\n\\[\n-1\\le\\sin x,\\tanh x\\le1,\\qquad\n\\cos x\\ge \\cos1.25\\approx0.315.\n\\]\nHence\n\\[\n0.315=2-1-1+\\cos1.25\\le\n2+\\sin x+\\cos y+\\tanh z\\le5,\n\\]\nand dividing by $4$ gives\n\\[\n0.07875\\le\\frac{2+\\sin x+\\cos y+\\tanh z}{4}\\le1.25 .\n\\]\nThus $F(I^{3})\\subseteq I^{3}$.  Together with (1.1) we conclude\n\\[\n-0.25\\le x_{j},y_{j},z_{j}\\le1.25 \\quad(j\\ge1).\n\\tag{1.2}\n\\]\n(This step is not essential for the contraction argument below, but it provides a convenient global bound.)\n\nStep 2.  Global contraction in the $\\|\\cdot\\|_{\\infty}$–norm.  \nLet $F:\\mathbb R^{3}\\to\\mathbb R^{3}$ denote the right–hand side of the recursion.  Its Jacobian is\n\\[\nDF(P)=\\frac14\n\\begin{pmatrix}\n\\cos x & -\\sin y & \\operatorname{sech}^{2} z\\\\\n\\operatorname{sech}^{2} x & \\cos y & -\\sin z\\\\\n-\\sin x & \\operatorname{sech}^{2} y & \\cos z\n\\end{pmatrix},\n\\qquad P=(x,y,z).\n\\]\nSince $|\\cos|\\le1$, $|\\sin|\\le1$ and $0<\\operatorname{sech}^{2}\\le1$, every entry of $DF(P)$ has absolute value $\\le\\frac14$.  Therefore the maximum row–sum norm satisfies\n\\[\n\\|DF(P)\\|_{\\infty}\\le\\frac14+\\frac14+\\frac14=\\frac34<1\n\\qquad(\\forall P\\in\\mathbb R^{3}),\n\\]\nso $F$ is a global contraction on $(\\mathbb R^{3},\\|\\cdot\\|_{\\infty})$ with Lipschitz constant $q=\\tfrac34$.\n\nStep 3.  Existence and uniqueness of a fixed point.  \nBy the Banach Fixed Point Theorem the contraction $F$ possesses a unique fixed point $P^{*}\\in\\mathbb R^{3}$, and every orbit satisfies\n\\[\n\\|P_{j}-P^{*}\\|_{\\infty}\\le q^{\\,j}\\,\\|P_{0}-P^{*}\\|_{\\infty}\\qquad(j\\ge0).\n\\tag{3.1}\n\\]\n\nStep 4.  Symmetry of the fixed point.  \nBecause $F$ is invariant under the cyclic permutation $(x,y,z)\\mapsto(y,z,x)$, uniqueness of the fixed point implies its invariance, hence\n\\[\nP^{*}=(L,L,L)\\quad\\text{for some }L\\in\\mathbb R.\n\\]\n\nStep 5.  Scalar characterisation of $L$.  \nSubstituting $x=y=z=L$ into the defining recurrence yields exactly\n\\[\nL=\\frac{2+\\sin L+\\cos L+\\tanh L}{4},\n\\]\nwhich is the equation stated in part (b).\n\nStep 6.  Uniqueness of the scalar root.  \nDefine $g(t)=\\frac{2+\\sin t+\\cos t+\\tanh t}{4}$.  Then\n\\[\n|g'(t)|=\\Bigl|\\tfrac14\\bigl(\\cos t-\\sin t+\\operatorname{sech}^{2}t\\bigr)\\Bigr|\n\\le\\tfrac14(1+1+1)=\\frac34<1\n\\qquad(\\forall t\\in\\mathbb R),\n\\]\nso $g$ is a contraction on $\\mathbb R$ with the same constant $q=\\tfrac34$.  Therefore it has a unique fixed point, namely $L$.\n\nStep 7.  Numerical evaluation to four decimals.  \nFor contractions the estimate\n\\[\n|L-t_{k}|\\le\\frac{1}{1-q}\\,|t_{k}-t_{k-1}|\n\\tag{7.1}\n\\]\nholds for the Picard iterates $t_{k+1}=g(t_{k})$.  Because $q=\\tfrac34$, the prefactor is\n\\[\n\\frac1{1-q}=\\frac1{1-\\frac34}=4.\n\\]\n\nStart with $t_{0}=1$ and iterate:\n\n\\[\n\\begin{array}{lcl}\nt_{1}=g(1)&=&1.0358418618,\\\\\nt_{2}=g(t_{1})&=&1.0366500807,\\\\\nt_{3}=g(t_{2})&=&1.0366274868.\n\\end{array}\n\\]\n\nFrom the last two iterates we have $|t_{3}-t_{2}|<2.3\\times10^{-5}$.  Inequality (7.1) therefore gives\n\\[\n|L-t_{3}|<4\\cdot2.3\\times10^{-5}=9.2\\times10^{-5}<10^{-4}.\n\\]\nThus $t_{3}$ is already accurate to four decimal places.  Rounding,\n\\[\n\\boxed{L\\approx1.0366}.\n\\]\nConsequently the sequence $(P_{j})$ issued from any starting point converges to $(1.0366,1.0366,1.0366)$, and the requested four–decimal value is $L=1.0366\\pm0.0001$.",
      "metadata": {
        "replaced_from": "harder_variant",
        "replacement_date": "2025-07-14T01:37:45.387701",
        "was_fixed": false,
        "difficulty_analysis": "• Dimensional escalation: the problem moves from a single real recursion to a coupled 3-dimensional nonlinear system; interactions among coordinates must be controlled simultaneously.  \n• Advanced tools: the solution demands Jacobian‐norm estimates, Banach’s Fixed Point Theorem in a normed vector space, and symmetry arguments to force diagonal convergence.  \n• Global (not local) analysis: unlike the original, which only needs an invariant interval, here a uniform Lipschitz bound on all of $\\mathbb R^{3}$ must be established.  \n• Uniqueness reasoning is subtler: cyclic symmetry plus uniqueness of the fixed point is used to deduce its diagonal form.  \n• Additional functions: the presence of $\\tanh$ alongside $\\sin$ and $\\cos$ requires combining trigonometric and hyperbolic estimates.  \nTogether these elements introduce multiple interacting concepts and a deeper theoretical framework, making the enhanced variant significantly harder than both the original problem and the easier kernel variant."
      }
    }
  },
  "checked": true,
  "problem_type": "proof"
}