proof writing - Trouble in proving that $|x|_p = max|x_j|$

We define p-norm in this way: $\|x\|_p = \{\sum ^N_j=_1|x_j|^p\}^ {1\over p}$
We know that It change to $\|x\|_p = \max|x_j| $ when $ p \to \infty $
How can I prove this ?

Answer

What we have to prove is that if $x_1,\dots,x_K$ are real numbers such that $0\leqslant x_k\lt 1$, then
$$\lim_{p\to \infty}\left(1+\sum_{k=1}^Kx_k^p\right)^{1/p}=1.$$
It's equivalent to show that
$$\lim_{p\to \infty}\frac 1p\log\left(1+\sum_{k=1}^Kx_k^p\right)=0.$$
It's the case, since for $t\geqslant 0$, $0\leqslant\log(1+t)\leqslant t$.