How do you prove the square root of a sum bound?

Question

For $x\in \mathbb{R}^n$, how do you prove the following bound: $$\sqrt{ \sum_{i=1}^{n} | x_i |^{2} } \leq \sum_{i=1}^{n} \sqrt{ | x_ i |^{2} }$$

Answer 1

In general if $a\geq 0$ and $b\geq 0$, then $(a+b)^2=a^2+b^2+2ab\ge a^2+b^2$

We proceed by induction. The base case is trivial. Suppose that the claim holds for $n-1$. Then $$ \begin{align} (|x_1|+\dots |x_n|)^2&\geq |x_n|^2+(|x_1|+\dotsb+|x_{n-1}|)^2\\ &\geq|x_n|^2+|x_1|^2+\dotsb+|x_{n-1}|^2 \end{align} $$ by the inductive hypothesis.