How to prove this involving sequences

Question

Let $a,b,c$ be $3$ positive numbers.

Show that $$\lim_{n\to\infty} (a^n+b^n+c^n)^{\frac{1}{n}}=\max\{a,b,c\}$$ Thank you for your time

How can I generalize this?

Answer 1

Hint

$$\Big(\max\{a,b,c\}\Big)^n<a^2+b^2+c^2<2\times \Big(\max\{a,b,c\}\Big)^n$$ for large enough $n$.

Answer 2

Suppose wlog that $\max\{a,b,c\}=a$. Then $$|\sqrt[n]{a^n+b^n+c^n}-a|\leqslant|\sqrt[n]{3a^n}-a|=|\sqrt[n]3a-a|\to0$$ since $\sqrt[n]3\to1$ as $n\to\infty$.