How is the interchange of the limit and the maximum valid at this point in Erwin Kreyszig?

Question

In 1.5-5 in Erwin Kreyszig's INTRODUCTORY FUNCTIONAL ANALYSIS WITH APPLICATIONS, the author shows completeness of the space $C[a,b]$ of all (real- or complex-valued) functions defined and continuous on the closed interval $[a,b]$ on the real line with the metric $$d(x,y) \colon= \max_{t\in[a,b]} |x(t) - y(t)|.$$

Now during the course of the proof, Kryszeg uses the following result: $$ \lim_{n\to\infty} \max_{t\in[a,b]} | x_m(t) - x_n(t)| = \max_{t\in[a,b]} | x_m(t) - \lim_{n\to\infty} x_n(t)| .$$

How is this sort of an interchange of the limit and the maximum valid? How to rigorously prove this?

To sum up, how to rigorously prove that $$\lim_{n\to\infty} \max_{t\in[a,b]} |x_n(t)| = \max_{t\in[a,b]} |\lim_{n\to\infty} x_n(t)|, $$ where $x_n \in C[a,b]$ for $n=1,2,3,\ldots$.

Answer 1

The core fact is to notice that the distance $d$ on $C[a,b]$ gives rise to a norm by the formula

$$\| f \| = d(f,0) = \max \limits _{t \in [a,b]} |f(t)| .$$

Naturally, then, the distance topology generated by $d$ and the norm topology generated by $\| \cdot \|$ will coincide (for instance, because they have the same balls) and we shall say that $f_n \overset {\| \cdot \|} \to f$ if and only if $\| f_n - f \| \to 0$ (i.e. if and only if $d(f_n, f) \to 0$, as expected).

The given question can be reformulated as:

if $f_n \overset {\| \cdot \|} \to f$, show that $\lim \| f_n - f \| = \| \lim (f_n - f) \|$ (i.e. that $\lim$ may jump inside the norm).

Of course, one may take $f=0$ (otherwise, just work with $g_n = f_n - f$), so the above becomes

if $f_n \overset {\| \cdot \|} \to 0$, show that $\lim \| f_n \| = \| \lim f_n \|$.

Now, it is clear that the function $f \mapsto \| f \|$ is continuous in the topology given by $\| \cdot \|$, thanks to the "reverse triangle inequality" $| \| f_n \| - \| f \| | \le \| f_n - f \|$ (note the modulus in the left-hand side). This continuity can be expressed as: if $f_n \overset {\| \cdot \|} \to f$, then $\| f_n \| \to \| f \|$. In particular, for $f=0$, if $f_n \overset {\| \cdot \|} \to 0$, then $\| f_n \| \to 0$ or, equivalently, $\lim \| f_n \| = 0 = \| 0 \| = \| \lim f_n \|$, which is precisely what was asked for.

(Incidentally, this topology is called the topology of uniform convergence and is one of the most important modes of convergence.)