Completeness of the space of bounded continuous functions

Statement

Let $X$ be a topological space and let $(Y, d_Y)$ be a complete metric space. Let $C_b(X, Y)$ be the set of continuous maps $f \colon X \to Y$ whose image $f(X)$ is bounded in $Y$. For $f, g \in C_b(X, Y)$, let $d_\infty(f, g) = \sup_{x \in X} d_Y(f(x), g(x))$, with $d_\infty(f, g) = 0$ when $X$ is empty. Then $(C_b(X, Y), d_\infty)$ is a complete metric space.

Remarks

For $Y = \mathbb{R}$, the metric $d_\infty$ is the uniform metric, and it comes from the sup norm. Proof sketch: a Cauchy sequence $(f_n)$ in $d_\infty$ is Cauchy at each point $x$, so it has a pointwise limit $f(x)$ in $Y$. The convergence is uniform, and a uniform limit of continuous maps is continuous. The limit is also bounded. For a compact space $X$, each continuous map to $\mathbb{R}$ is bounded, so $C(X, \mathbb{R})$ is complete with the sup metric.

Explanations

L1 human

If continuous functions get closer and closer to each other uniformly, then they converge uniformly to a function. The key point is that this limit function is again continuous, so the limit stays in the space.