Question: Prove that $(B, \|-\|_{\infty})$ is complete. B the set of bounded real valued functions on [0,1] which are pointwise limit of continuous functions on [0,1].
Context: Old exam problem I'm using to study. Real Analysis by Carothers.
I've attempted to avoid a direct proof from the definition using Cauchy sequences by appealing to the fact that ($B_{\infty}, \|-\|_{\infty}$) is a normed space where $B_{\infty}$ is the set of bounded real valued functions on $\mathbb{R}$ that are pointwise limits of continuous functions. So $B \in B_{\infty}$ and by a Theorem a normed spacve is complete if and only if every absolutely summable series in $B$ is summable.
I'm having trouble making a solid case for proving the condition in the last part and also how to say that $B$ is indeed a normed space.
Thank you in advance.