Completeness of $B(X,R)$

Question

I'm trying to show $B(X,\mathbb R)$ the set of all bounded functions from a metric space $X$ to $\mathbb R$ is a complete metric space with respect to supremum metric, if $X$ is complete.

So I took a Cauchy sequence in $B(X,\mathbb R)$ and I'm trying to show it is convergent. My question is do we show it is pointwise convergent or uniformly?

Answer 1

I'm assuming that $d(f,g)=\sup\{f(x)-g(x):x\in X\}$. If so, convergence with respect to this metric is uniform convergence.

Answer 2

Think about $ f_{n}(x)$ as Cauchy sequence and since your range of the functions is in $\mathbb{R}$ so they are convergent with uniform convergence because $\mathbb{R}$ is complete.

Answer 3

Well, when you say by Supreme metric space, it does mean uninformative convergence. Of course uninformative convergence is stronger than point-wise convergence.The norm of $B(X)$ is $ \Vert f\Vert_{\infty} = \sup\limits_{x\in X} \vert f(x)\vert. $ So by your assumption, it is uninformative. For more information, please refer to the "Principles of real analysis, 3rd ed. C 1998" chapter 2 section 9 by "Aliprantis".