The problem
Let $CBT(X,Y)$ be the set of continuous, bounded functions that map topological space $(X, \tau)$ to a complete metric space $(Y, \rho)$. Define a metric on $CBT$ as follows: $\rho_\infty(f, g) := \underset{x \in X}{\sup}\rho(f(x), g(x))$.
It is needed to prove that $(CBT(X, Y),\ \rho_\infty)$ is a complete metric space.
My ideas
I am quite sure I've seen something similar before, but not this abstract. It is probably the fact that topology $\tau$ can be arbitrary that is confusing me.
Having a fundamental sequence of functions from $CBT$ it is not difficult to prove it converges pointwise. I would define the presumed limit function $f$ as $f(x) := \lim\limits_{n \to \infty}f_n(x)$.
Now I believe I need to prove $\{f_n\}$ converges to $f$ in $\rho_\infty$, and that $f$ is itself continuous and bounded.
I am interested what is the usual way to prove this kind of things and whether my idea could work?