Answer by user147263 for Sequence of linear functionals

I think the solution suggested by Jochen is the intended one for this exercise. By the closed graph theorem, it suffices to prove that if $x_k\to x$ and $\Phi x_k\to y$, then $\Phi(x)=y$. The equality can be checked coordinate-wide, and for each coordinate (say $n$th) it follows from the continuity of $\phi_n$.