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Let $B = (B(X, \mathbb{R}), d_u)$ be the set of all bounded functions from the metric space $X$ into $\mathbb{R}$.

Let $(f_n(x))$ be a Cauchy sequence in $B$.

Is the following statement valid -

As $(f_n(x))$ is a Cauchy sequence it exhibits pointwise convergence in that for all $x \in X$ $f_n(x) \to f(x)$ as $n \to \inf$?

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1 Answer 1

up vote 1 down vote accepted

If you equip $B$ with the sup norm, then this is true, since $|f_{n}(x) - f(x)| \leq ||f_n - f||_{\sup} \to 0$. Of course, the existence of such a function $f$ is not immediate - it follows from the fact that $B$ is complete, which is a standard real-analysis theorem.

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