State and prove the Closed Graph Theorem. [You may assume any version of the Baire Category Theorem provided it is clearly stated. If you use any other result from the course, then you must prove it.]

Let $X$ be a closed subspace of $\ell_{\infty}$ such that $X$ is also a subset of $\ell_{1}$. Show that the left-shift $L: X \rightarrow \ell_{1}$, given by $L\left(x_{1}, x_{2}, x_{3}, \ldots\right)=\left(x_{2}, x_{3}, \ldots\right)$, is bounded when $X$ is equipped with the sup-norm.