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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.

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  • Last sentence in this answer. The Op already knows that a uniform limit of continuous functions is again continuous. So it boils down to 1. show the space of bounded functions is complete, and 2. note that $C_b$ is a closed subspace of that. – Daniel Fischer Oct 19 '16 at 13:13

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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.

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