I was reading a proof in my notes that we went over in class. It says let ${f_n}$ be Cauchy in $C(X)$ where $C(X)$ is the space of bounded and continuous functions. This translates to ${f_n}$ being uniformly Cauchy in the sense of Rudin's Theorem 7.8.
No justification is given in my notes, and I don't understand how $f_n$ being Cauchy in $C(X)$ translates to $f_n$ being uniformly Cauchy.
Any help would be much appreciated.
The space of bounded continuous functions is (presumably) equipped with the sup norm, so given $\epsilon > 0$ there is $N$ such that \begin{equation} \sup_{x \in X} |f_n (x) - f_m (x)| < \epsilon \text{ for all } m, n > N. \label{1} \end{equation} For all $x \in X$, we have \begin{equation*} |f_n (x) - f_m (x)| \leq \sup_{x \in X} |f_n (x) - f_m (x)| < \epsilon \text{ for all } m, n > N. \end{equation*} Since $N$ is independent of $x$, this is exactly what we mean by uniformly Cauchy.