Corollary to Markov's Inequality

Question

Here is a particular case of Markov's Inequality that I failed to prove:

Let $X$ be a non-negative integer-valued random variable with $\mathbb{E}(X)\leq m$ then $$\mathbb{P}(X=0)\geq 1-m$$

Does anyone has an idea how to prove this? Thank you.

Answer 1

$$m\geq\mathsf EX=\sum_{n=0}^{\infty}n\mathsf P(X=n)\geq\sum_{n=1}^{\infty}\mathsf P(X=n)=\mathsf P(X\geq1)=1-\mathsf P(X=0)$$