Confusion about covering number
Since you've tagged machine learning, I'll address this setting. Indeed, for the usual $\ell_p$ norms on $\mathbb{R}^n$, we have $||x||_p\le ||x||_q$ for $p&gt;q$. However, in machine learning applications, these are normalized norms (see, e.g., Mendelson, A few notes on Statistical Learning Theory
https://people.eecs.berkeley.edu/~jordan/courses/281B-spring04/readings/mendelson.ps
). 
Let $D$ be a distribution on $\{1,\ldots,n\}$. Then the corresponding $\ell_p(D)$ norm is defined by
$$
||x||_{\ell_p(D)}^p = \sum_{i=1}^n D(i)|x_i|^p.
$$
You can verify that for these norms, the inequality goes the other way:
$||x||_{\ell_p(D)}\ge ||x||_{\ell_q(D)}$ for $p&gt;q$. This explains the covering number inequality as well.