Let us consider a random variable $X$ which is normally distributed with mean $0$ and variance $1$, that is:
$$X\sim\mathcal{N}\left(0, 1\right)\tag{1}$$
Could I be sure that for some positive integer $h$ it holds true that:
$$\mathbb{P}\left(X>h\right)>0$$?
That is, equivalently, could I be sure of the fact that it is not possible that $\mathbb{P}\left(X>h\right)=0$? If so, why? Has one to think of standard normal complementary cumulative distribution function?
I was thinking that knowing $(1)$ cannot make me sure of the fact that almost surely the random variable $X$ does not take values greater than $h$, since $h$ is just some positive integer, and it could be arbitrarily big. I am struggling with that, could you please help me understand?