The problem is as follows: Let S be a sample space, and let X and X' be random variables such that X(s) ≥ X'(s) for all s ∈ S. Prove that for any real constant t, Pr{X ≥ t}≥Pr{X'≥ t}.
I can understand this intuitively but don't know how to formulate a proof.
Guide:
Prove that $$\{ s|X'(s) \ge t\} \subseteq \{ s|X(s) \ge t\}$$
That is equivalent to $\forall r \in \{ s|X'(s) \ge t\}$, we have $r \in \{ s|X(s) \ge t\}$