How to prove that $\lim_{n\to\infty} \sqrt[n]{a^n+b^n}=\operatorname{max}(a,b)$?

Assume that $a \geq b$. Else, relabel the numbers $a \leftrightarrow b$. This being understood, we have $a = \max(a,b)$. Then $$a =\sqrt[n]{a^n} \leq \sqrt{a^n+b^n} \leq \sqrt{a^n + a^n} = \sqrt[n]{2} a. $$Now apply $\lim_{n \to +\infty}$ on everything, noting that $\sqrt[n]{2} \to 1$. It follows from the squeeze theorem that $$\lim_{n\to +\infty}\sqrt[n]{a^n+b^n} = a,$$ as wanted.