\section{PROOFS}\label{sec:Proofs}
\input{sections/Supplementary/Proof of Lem_Norm_ineqaulity}
\input{sections/Supplementary/Proof of Main Prop}
\input{sections/Supplementary/Proof of Thm1}
\input{sections/Supplementary/Proof of Thm2}
\input{sections/Supplementary/Proof of Lem_relating norms}
\input{sections/Supplementary/Proof of Prop_finite_opt}