Triangle Inequality for $\|x\|_{\infty}$

Question

I have to show the triangle inequality for $\|x\|_{\infty}$. I'm not sure, if estimate is correct. To show: $\|x+y\|_{\infty} \le \|x\|_{\infty}+\|y\|_{\infty}$

Let $x \in \mathbb{R}^n$ and $\|x\|_{\infty}=\max\{|x_1|,\dots,|x_n|\}$:

$\|x+y\|_{\infty}=\max\{|x_1+y_1|,\dots,|x_n+y_n|\} \le \max\{|x_1|+|y_1|,\dots,|x_n|+|y_n|\} \le \max\{|x_1|,\dots,|x_n|\} +\max\{|y_1|,\dots,|y_n|\} \, \square$

Unfortunately I haven't any other appropriate ideas. Thanks for help!

Answer 1

The proof is fine. But if you do not find it convincing yet this might mean you should add more details.

For example, you could justify the last inequality a bit more by saying that for each $j$ one has $|x_j| \le \max\{|x_1|,...,|x_n|\}$ and $|y_j| \le \max\{|y_1|,...,|y_n|\} $ and thus $|x_j|+|y_j| \le \max\{|x_1|,...,|x_n|\} +\max\{|y_1|,...,|y_n|\} $.