Some proof by induction

Question

I want to prove $|x_1\cdot x_2\cdots x_n| \leq |x_1|^n+\cdots +|x_n|^n$ for all $n\in\mathbb{N}$ and $(x_1,\ldots,x_n)^T \in \mathbb{R}^n$. For $n=1$ and $n=2$ its obvious. Assuming it holds for one $n$ we obtain

$$|x_1\cdot x_2\cdots x_n\cdot x_{n+1}| \leq (|x_1|^n+\cdots +|x_n|^n)|x_{n+1}|.$$

From here I couldn't proceed.

Answer 1

Hint: pick the largest of the $|x_i|$s. For instance, for $k=2$ we have $$|ab|\leqslant \max\{|a|,|b|\}^2 \leqslant |a|^2+|b|^2.$$ Do similarly for $k=n$.