Let $X$ be a Banach space and $M:X\rightarrow X$ a linear map. Suppose that there exists a dense subset $S\subseteq X^{\prime}$ of the dual such that for each sequence $(x_{n})_{n\in\mathbb{N}}$ tending to zero in $X$ and $l\in S$, we have
\begin{align} \lim_{n\rightarrow\infty}l(Mx_n) =0. \end{align}
Show that $M$ is bounded.
I tried to show that $M$ is continuous at $0$ but did not manage to sort it out. Maybe closed graph theorem?