$p$-norm on $\mathbb{R}^n$ question

Question

How I can show that $$\lim_{p \to \infty} \|x\|_{p} = \max\{|x_1|, \; |x_2|, \; \cdot ,\; |x_n|\}$$ if $\mathbb{R}^n$ has the p-norm? $p > 1$ of course.

Has anyone done this or know how to? I'm reading about p-norms and this was states, but not shown in the text.

Answer 1

Note that if $\lvert x_k\rvert=\max\{\lvert x_1\rvert,\ldots,\lvert x_n\rvert\}$, then for all $p$ $$ \lvert x_k^p\rvert\le \|x\|_p^p\le n\lvert x_k\rvert^p, $$ and hence $$ \lvert x_k\rvert\le \|x\|_p\le n^{1/p}\lvert x_k\rvert. $$ Thus $$ \lim_{p\to\infty}\|x\|_p=\lvert x_k\rvert=\max\{\lvert x_1\rvert,\ldots,\lvert x_n\rvert\}. $$

Answer 2

Hint: $$ \|x\|_\infty^p\le \| x \|_p^p \le n\|x\|_\infty^p $$