For any non-zero natural integer $n$, we set
$$\omega(n) = \operatorname{Card}\{p \text{ prime} : p \mid n\} = \sum_{\substack{p \mid n \\ p \text{ prime}}} 1.$$
Prove, using a change of order of summation, that $\frac{1}{x} \sum_{n \leqslant x} \omega(n) \underset{x \rightarrow +\infty}{=} \ln_2(x) + O(1)$.