Let $X$ be a non-negative integer-valued random variable such that $0<\mathbb{E}\left(X^{2}\right)<\infty$.

Prove that

$$\frac{\mathbb{E}(X)^{2}}{\mathbb{E}\left(X^{2}\right)} \leqslant \mathbb{P}(X>0) \leqslant \mathbb{E}(X)$$

[You may use any standard inequality.]