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I am having trouble proving this.

Prove that lim$_{p→∞}$ ||v ||$_p$ = ||v||$_∞$ for any v $\in$ $R^2$.

My attempt:

We have that ||v ||$_p$ = p (∑||v$_i$||p)^1/2 from the interval of i=1 to infinity,which equals to ||v ||$_p$ = (|v$_1$| + |v$_2$| ... + |v$_n$|) but from here on how can I show that it is equal to ||v||$_∞$. It seems very obvious it is, but how can I show it formally?

Note: Sorry for my coding format. I am very bad at it.

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1 Answer 1

up vote 1 down vote accepted

HINT

Note that $$n^{1/p} \Vert v \Vert_{\infty} \geq \Vert v \Vert_p \geq \Vert v \Vert_{\infty} \,\,\, (\text{Why?})$$where $v \in \mathbb{R}^n$.

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Because if p -> ∞ then all ∥v∥$_p$ = ∥v∥$_∞$ ? –  diimension Oct 1 '12 at 21:09
    
@diimension Yes. –  user17762 Oct 1 '12 at 21:10
    
Wow, so that keep completes the proof? That simple? –  diimension Oct 1 '12 at 21:14

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