Let $\ell^2$ be the set of real number sequences $\{a\_n\}$ such that $\sum a\_n^2 <\infty$. Let $\langle a\_n,y\rangle \rightarrow \langle a,y\rangle$ for some $a\in \ell^2$ and for all $y\in \ell^2$. 

Would you help me to prove that $\{\| a\_n\|\}$ bounded and for a fixed $i$, $a\_i^{(n)} \to a^{(n)}$ where $x^{(n)}$ denote the $n$-component of $x$.