Let $X \in \text{Random}(\mathbb{R})$ be a D3161: Random real number such that

(i)	$[a, b] \subset \mathbb{R}$ is a D544: Closed real interval
(ii)	$$\begin{equation} \mathbb{P}(X \in [a, b]) = 1 \end{equation}$$

Since $X \in [a, b]$ almost surely, result R4778: Stieltjes integral calculus expression for probability that a bounded random real number takes value on the bounding interval shows that $\int^b_a \, d F(x) = 1$. Thus, we have $$\begin{equation} \mathbb{E} X = \int^b_a x \, d F(x) \leq b \int^b_a \, d F(x) = b \end{equation}$$ and $$\begin{equation} \mathbb{E} X = \int^b_a x \, d F(x) \geq a \int^b_a \, d F(x) = a \end{equation}$$ This is what was required to be shown. $\square$