The exercise goes as follows: Show that $||x||_\infty \le ||x||_2 \le ||x||_1$. Which I believe I proved--
W.L.O.G., suppose $max |x_i| = |x_1|\ \ |x_k| = \sqrt{x_k^2} \le \sqrt{x_1^2 +...+x_n^2}\ \ \therefore\, ||x||_\infty \le ||x||_2$. Note that $|x_1|+...+|x_n| = \sqrt{x_1^2} + ... + \sqrt{ x_n^2} \ge \sqrt{x_1^2 +...+x_n^2}\ \ \therefore\ ||x||_1 \ge ||x||_2$.
But it then continues to say "Also check that $||x||_1 \le n||x||_\infty$ and $||x||_1 \le \sqrt{n}||x||_2$." This part confuses me, as if $n = 1$ then we are still left with inequality mentioned above. I'm assuming $n$ is equal to the $n$ in $\mathbb{R}^n$ (since it does not specify it otherwise). I would appreciate if someone were to hint me as to what I am missing.
$n$ is the $\color{blue}n$ in $\mathbb{R}^{\color{blue}{n}}$.
For your first question:$$\sum_{i=1}^n |x_i| \le \sum_{i=1}^n \max_j |x_j|.$$
Hint for the second question: Cauchy-Schwarz.