Graduate Limits and Supremum: Is It True?

[Mathvsphysics]

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 $\sup_{s\in S} <a,s><+\infty$

 

[PeroK]

You need to use two hashes (or two dollars) for Latex delimiters.

 

[fresh_42]

Corrected.
Here is explained how you can type formulas on PF: https://www.physicsforums.com/help/latexhelp/

 

[FactChecker]

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?