Codomain of function complete implies function space complete

Question

Let $(X,d_X)$ and $(Y,d_Y)$ be two metric spaces.

Let $(C_b(X,Y), d_C)$ be a function space with the metric $$d_C(f,g):=\sup\limits_{x\in X} d_Y(f(x), g(x)).$$ Prove that if $(Y,d_Y)$ is complete then $(C_b(X,Y), d_C)$ is complete.

Unfortunately, I have no idea how to approach this problem. I can think of something like this:

Let $(y_k)_k\subset Y$ be a Cauchy sequence converging to $y\in Y$. Then there exists some $f\in C_b(X,Y)$ such that $f(x_k)=y_k$ for all $k\in \mathbb N$. Since $f$ is continuous, $\lim\limits_{k\to\infty} d_Y(f(x_k), y)=0$.

But how do I go from here? As far as I understand, what needs to be proved is that any Cauchy sequence of functions in $C_b(X,Y)$ converges in $C_b(X,Y)$, that is, that the limit is bounded and continuous. But how do we devise such a sequence of functions?

Answer 1

If you want to prove that $C_b$ is complete, you should probably start with: let $(f_n)\subset C_b$ be a Cauchy sequence of functions.

You want to show that the sequence is convergent, and that the limit lies in $C_b$. For now, let's call that limit $f$ (we still don't know it exists, but this is more of a 'how do we approach solving this problem' type of answer). What should $f$ look like?

Well, certainly, $f_n(x)$ should converge to $f(x)$ -- pointwise convergence is weaker than uniform convergence.

Step $(1)$: Show that $\big(f_n(x)\big)_{n\in\mathbb{N}}\subset Y$ is Cauchy.

Well, now things are looking better: it's clear where the completeness of $Y$ comes into play. It follows that for each $x\in X$, $\big(f_n(x)\big)$ is converges to some limit in $Y$ which we'll define as $f(x)$.

Well, this defines $f$ everywhere. Are we done then? No, we still need to show that $f$ lies in $C_b$. In other words, we need to show that $f$ is continuous.

Here is where the norm of $X$ will come into play.

Step $(2)$: Show that $f$ is continuous.

By the definition, we need to show that

$$\forall x_0\in X,\, \forall \epsilon>0,\,\exists\delta>0,\,\forall x_1\in X \text{ with $d_X(x_0,x_1)<\delta$,}\, d_Y\big(f(x_0),f(x_1)\big)<\epsilon$$

Now, we may use the triangle inequality to get something like

$$d_Y\big(f(x_0),f(x_1)\big)\leq d_Y\big(f(x_0),f_n(x_0)\big)+d_Y\big(f_n(x_0),f_n(x_1)\big)+d_Y\big(f_n(x_1),f(x_1)\big)$$

Do you think you can finish on your own?