My textbook says that the Lp-norm converges to the Chebyshev norm. This seems intuitive, but I fail to prove it.

Thus, my question is, how do I prove that $\displaystyle \lim_{p\rightarrow \infty} (|x_1|^p+|x_2|^p+\dots+|x_n|^p)^{1/p} = \max(|x_1|,\dots,|x_n|)$