Proof of $L^p $ norm

Question

Can someone show me how to proof this ? $$\sqrt[p]{\sum_i {a_i}^p}\to \max a_i \quad\text{ if }p\to\infty.$$

Answer 1

Assuming $\forall i, a_i \le 0$

We have :$$\sum_i a_i^p \sim k \max_i a_i^p$$ where $k$ is the number of times the number $\max_i a_i$ appears in the sum.

Thus $$\sqrt[p]{\sum_i a_i^p}\sim \sqrt[p]k\max_i{a_i} \sim \max_i{a_i}$$

(since $k\ge1$)