exchange max and limit Let $\lim_{n \rightarrow \infty} a_n = a$ and $\lim_{n \rightarrow \infty} b_n
= b$ exist, then is it true that $\lim_{n \rightarrow \infty} \max \{ a_n, b_n \} = \max \{ a, b \}$?
I couldn't find this on wiki, but it seems correct. Here's my proof, can somebody check it?
Suppose $a > b$. Let $\epsilon^{\prime} > 0$. Take $0 < \epsilon < \min \{
\epsilon^{\prime}, (a - b) / 2 \}$. Then there is an $N$ s.t. $n > N
\Rightarrow | a - a_n | < \epsilon \wedge | b - b_n | < \epsilon \Rightarrow
a_n - b_n > (a - \epsilon) - (b + \epsilon) > 0 \Rightarrow | \max \{ a_n, b_n
\} - a | = | a_n - a | < \epsilon < \epsilon^{\prime}$.
Suppose $c := a = b$. Let $\epsilon > 0$. Then there is an $N$ s.t. $n >
N \Rightarrow | c - a_n | < \epsilon \wedge | c - b_n | < \epsilon \Rightarrow
| c - \max \{ a_n, b_n \} | < \epsilon$.
 A: For posterity: On the set of real numbers, $\max$ and $\min$ are given by
$$
\max(a, b) = \frac{a + b + |b - a|}{2},\qquad
\min(a, b) = \frac{a + b - |b - a|}{2}.
$$
(Proof: "Go to the midpoint $\frac{1}{2}(a + b)$, then step to the right or left by half the distance between the numbers, $\frac{1}{2}|b - a|$.")
These functions are (almost obviously) continuous as functions of two variables, so
$$
\lim_{n \to \infty} \max(a_{n}, b_{n})
  = \max\bigl(\lim_{n \to \infty} a_{n}, \lim_{n \to \infty} b_{n}\bigr)
  = \max(a, b),
$$
and similarly for $\min$.
In fact, continuity guarantees stronger "double limit" assertions:
$$
\max(a, b)
  = \lim_{m \to \infty} \lim_{n \to \infty} \max(a_{m}, b_{n})
  = \lim_{n \to \infty} \lim_{m \to \infty} \max(a_{m}, b_{n}),
$$
etc.
A: A kind of overkill: fix $K$ compact (in this case $K=\{1,2\}$) and consider the Banach space $(\mathcal{C}(K),\|\cdot\|_\infty)$. Since the norm is continuous, then $\lim_n \|f_n\|_\infty=\|\lim_n f_n\|_\infty$ for each sequence $(f_n)$ in $\mathcal{C}(K)$, provided the limits exist.
To conclude the proof, use that $x_n \to x$ iff $x_n^+ \to x^+$ and $x_n^- \to x^-$, where $a=a^+-a^-$ is the unique decomposition into positive and negative part.
