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 $n \to \inf$?