Recall Theorem 7.25 of Rudin's Principles of Mathematical Analysis:
If $K$ is compact and $f_n \in \mathscr{C}(K)$ for $n \in \mathbb{N}$, and if $\{f_n\}$ is pointwise bounded and equicontinuous on $K$, then:
(a) $\{f_n\}$ is uniformly bounded on $K$.
(b) $\{f_n\}$ contains a uniformly convergent subsequence.
Here is the beginning of Rudin's proof of (a), suitably rewritten for my own intuition:
Let $\epsilon > 0 $ and, by equicontinuity, pick $\delta > 0$ such that $d(x,y) < \delta \implies |f_n(x) – f_n(y)| < \epsilon$ for all $n$. Cover $K$ with $\delta$-balls, and find a finite subcover $P$ with centers $p_1, \dots, p_r$. $f_n$ is pointwise bounded, so $f_n(p_i) < M_i$ for all $n$. Let $M = \max\{M_1,\dots,M_r\}$. Then $|f(x)| < M + \epsilon$. (Note the importance of $\epsilon$, while $f(p_i) < M$, this is only for $p_i \in P$. We get the bound $M + \epsilon$ by equicontinuity, since all $x \in B(p_i, \delta)$ for some $i \in J_n$.
For the most part, this is clear, however, he chose to wrote $|f(x)| < M + \epsilon$ rather than $|f_n(x)| < M + \epsilon$. Is this simply abuse of notation, or am I missing something elementary?
Best Answer
This is a typo that is fixed in newer versions of the 3rd edition. You are correct that it should read $$ |f_n(x)|<M+\varepsilon. $$