I have a question whether given a random variable $X$ such that $P(X ≥ 0) = 1$ and $P(X > 0) > 0$ it is true that $E[X] > 0$? It was mentioned as a remark during a lecture without proof, but I don't see a way to prove it from the definitions (and without using Markov's inequality, since it wasn't introduced yet in my course, and I'm mostly interested in building intuitions/knowledge regarding those fundamental notions).
For simplicity: I'm only concerned about a discrete case. I would appreciate any help!