Let $C_b(\mathbb{R})$be the space of all bounded continuous functions on $\mathbb{R}$, normed with $$\|f\|= \sup_{x\in \mathbb{R}}|f(x)|$$
Show that this space is complete.
Complete mean that all Cauchy sequences converges.
So if we have an Cauchy sequence $(f_n)$, define $f(x) = \lim_{n \rightarrow \infty} f_n(x)$, we much show that $f\in C_b(\mathbb{R})$ and $\|f – f_n\| \leq \epsilon.$
How can I proceed? If I take $f(x) = f_n(x) + (f(x) – f_n(x))$ and show that the last parentheses goes too zero? But then I end up with $$|\lim_{m \rightarrow \infty} f_m(x) – f_n(x)|$$
how can I proceed from here? move out the limes? is that possible? how do one reason?
Best Answer
Sketch (for a more general result):
Step 1: Prove that $\ell_\infty(X)$, the set of bounded functions on $X$, is complete for any set $X$. Sketch/Hint: Uniformly cauchy implies pointwise cauchy which implies pointwise convergent. Then show that the pointwise limit is bounded. Then show that the uniform limit exists and equals the pointwise one.
Step 2: Prove that if $X$ is a metric space (or more generally a Hausdorff topological space) and if $(f_n)$ is a sequence of functions in $\ell_\infty(X)$ with each $f_n$ continuous at $x_0 \in X$ such that $f_n$ converges uniformly to a function $f$, then $f$ is continuous at $x_0$.
Step 3: Conclude that $C_b(X)$ is closed in $\ell_\infty(X)$ and is thus complete.