A Limits and Supremum: Is It True?

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We have ##a_n## converges in norm to ##a## and a set ##S## such that for all ##n\ge 0##
$$\sup_{s\in S} <a_n,s><+\infty .$$ Is it true that ##\sup_{s\in S} <a,s><+\infty##
 
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You need to use two hashes (or two dollars) for Latex delimiters.
 
Is this a homework-type of problem? There is a format for those and you must show work before we can give hints.
Suppose ##M \in R## is such that ##sup<a_n,s> \lt M##. Also, suppose ##\epsilon \gt 0## and ##m\in N## are such that ##<a_n,a> \lt \epsilon## ##\forall n\gt m##. What can you say then?
 
A sphere as topological manifold can be defined by gluing together the boundary of two disk. Basically one starts assigning each disk the subspace topology from ##\mathbb R^2## and then taking the quotient topology obtained by gluing their boundaries. Starting from the above definition of 2-sphere as topological manifold, shows that it is homeomorphic to the "embedded" sphere understood as subset of ##\mathbb R^3## in the subspace topology.

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