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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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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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    Wow, so that keep completes the proof? That simple?2012-10-01