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?