Understanding the proof that $C_b(\mathbb{R})$ is complete.

Question

There is something I don't get from the first anwswer to this question:

Space of bounded continuous functions is complete

It's proved for limited functions, but the question is for limited continuous functions, and I can't see where continuity was used in the proof. Can someone explain to me why proving that $B(x)$ is complete imply $C_b(\mathbb{R})$ complete?

Thanks.

Answer 1

This is a consequence of the fact that the norm $\|\cdot\|_\infty $ is the norm of the uniform convergence. It is known that if a sequence of continuous functions has a limit in the sense of the uniform convergence, that limit is a continuous function.