Proving norm inequality There is a brief proof in my textbook that I have one question about.
We are supposed to prove that $||x||_{1} \leq n||x||_{\infty}$ for $x \in \mathbb{R}^n$
The book writes the following:
$||x||_{1} = \sum_{i=1}^{n} |x_i| \leq \sum_{i=1}^{n}\{\max_{1 \leq j \leq n} |x_j| \} \leq \sum_{i=1}^{n} ||x||_{\infty} = n||x||_{\infty}$
The one thing I don't quite follow is when the book writes:
$\sum_{i=1}^{n}\{\max_{1 \leq j \leq n} |x_j| \} \leq \sum_{i=1}^{n} ||x||_{\infty}$.  In my book the definition of $||x||_{\infty}$ is given as:
$$||x||_{\infty} = \max_{1 \leq j \leq n}|x_j|$$
So shouldn't the inequality $\sum_{i=1}^{n}\{\max_{1 \leq j \leq n} |x_j| \} \leq \sum_{i=1}^{n} ||x||_{\infty}$  actually be an equality?
I would really appreciate it if someone could explain this to me!

It would also be correct with equality, but it’s not incorrect as it stands. After all, if $a=b$, then it’s also true that $a\le b$, and the inequality is all that’s actually needed here.
