Is $|x_1|+|x_2|+...+|x_n| \leq \sqrt{x_1^2+x_2^2+...+x_n^2}$?

Question

$$|x_1|+|x_2|+...+|x_n| \leq \sqrt{x_1^2+x_2^2+...+x_n^2}$$

This inequality is wrong has

$$\sqrt{x_1^2+x_2^2+...+x_n^2}\leq \sqrt{(x_1+x_2+...+x_n)^2}\leq|x_1+x_2+...+x_n|\leq |x_1|+|x_2|+...+|x_n|$$

But where in the process I got it wrong?

$$|x_1|+|x_2|+...+|x_n|=\sqrt{x_1^2}+\sqrt{x_2^2}+...+\sqrt{x_n^2}\leq \sqrt{x_1^2+x_2^2+...+x_n^2}$$

Answer 1

Try $x_i=1$.

You'll get $$n\leq\sqrt{n},$$ which is wrong for all $n>1$.

Answer 2

This part is not correct:

$$ \sqrt{x_1^2+x_2^2+...+x_n^2}\leq \sqrt{(x_1+x_2+...+x_n)^2}$$

Let $x_1 = 3, x_2 = -4$. Then $\sqrt{3^2 + (-4)^2} = 5 > 1 = \sqrt{(3 + (-4))^2}$