Let $X$ be a random variable such that $\mathbb{P}(X=0)<1$ and $\mathbb{E}X=-\varepsilon$, where $\varepsilon>0$.
Can you show that $\mathbb{P}(X>0)>0$ ?
It is natural to expect that the inequality may hold only for sufficiently small and positive $\varepsilon$.
Update. Let's add that $$\lim_{\varepsilon\to0}\mathbb{E}X^2>0.$$
No, it is not true : with a random variable such as $P(X=0)=1-\varepsilon< 1$ and $P(X=-1)=\varepsilon$, you will have $P(X>0)=0$ and $\mathbb{E}X=-\varepsilon$.