I'm reading Real and Functional Analysis by Lang. For normed vector spaces (NVS) $E,F$, he defines $L(E,F)$ as the space of continuous linear functions from $E$ to $F$. One proposition is that if $F$ is complete, then so is $L(E,F)$.
I think I have finished the proof. I have constructed the function $\lambda$ corresponding to the Cauchy sequence $(\lambda_n)$ in $L(E,F)$ by letting $\lim_{n\to\infty} \lambda_n(x) = \lambda(x)$. I have subsequently shown that $\lambda \in L(E,F)$. Is that enough?
Edit: showing convergence
We know $|\lambda_n(x) - \lambda(x)| \to 0$ for all $x\in E$. $|\lambda_n - \lambda| = \sup \{ |\lambda_n(x) - \lambda(x)| : |x|= 1 \}$. As $n\to \infty$, this set converges to $\{0\}$ so the supremum converges to $0$. Is that a correct line of thinking?