This is something that I thought I knew how to do until I actually sat down to do it.Prove that $\displaystyle ||x||_p \rightarrow ||x||_{\infty}$ as $\displaystyle p \rightarrow \infty$.

So I have to show that:

$\displaystyle \lim_{p \rightarrow \infty} \left( \sum_{i=1}^n |x_i|^p \right)^{\frac{1}{p}}= \max \{|x_1|,...,|x_n| \}$

I tried doing a proof by induction, but it didn't really get me anywhere. Can anyone give me a hint or tip?