Let $X$ be a Banach space. Prove that a linear map $M\colon X\mapsto \ell^p, \; p\geqslant 1$ is continuous iff for every sequence $(x_k)$ that converges in $X$ to $x \in X$, we have that the $n$-th term of the sequence $Mx_k$ converges to the $n$-th term of $Mx$ for all $n$.
My try: $(\Rightarrow)$ If $M$ is continuous $\|Mx_n - Mx\| \leq \epsilon$ hence $|(Mx_n)_i - (Mx)_i| \leq \epsilon$
$(\Leftarrow)$ $y\in \ell^p \Rightarrow \|y\|_p < \infty \Rightarrow \exists N : \|y\|_p^p = \sum_{i = 1}^N |y_i|^p + \underbrace{\sum_{i = N+1}^\infty |y_i|^p}_{\leq \epsilon}$
So we can find an $N$ such that $\|Mx-y\|_p^p = \underbrace{\sum_{i = 1}^N |(Mx)_i - y_i|^p}_{\leq N\epsilon} + \underbrace{\sum_{i = N+1}^\infty |(Mx)_i - y_i|^p}_{\leq 2\epsilon} \leq (N+2)\epsilon.$ Hence by closed graph we are done. Is this correct? What happens when $p = \infty$ ?