general topology - Show that $lim_{p→∞} ||x||_p = ||x||_∞$

For any $x ∈ \mathbb{R}^n$ and $p ≥ 1$, define $$||x||_p = \left(\sum_{i=1}^n|x_i|^p\right)^{\frac1p},||x||_∞ = \underset{1≤i≤n}{\max}|x_i| $$
Show that $$ \underset{p→∞}{\lim} ||x||_p = ||x||_∞$$

I understand this in theory, that if $x_i < x_j$ then as they are both raised to the power of ∞ then $x_i$ will become arbitrarily small relative to $x_j$, and so their sum to the power of $\frac1∞$ is $\approx x_j$. But I don't know how to go about proving this formally.