By definition of expectation, we have

\[

\begin{aligned}

E X &=\sum_{k \in X(\Omega)} k P(X=k) \\

&=\sum_{k \in X(\Omega) \text { st. } k \geq a} k P(X=k)+\sum_{k \in X(\Omega) \text { s.t. } k<a} k P(X=k) \\

& \geq \sum_{k \in X(\Omega) \text { s.t. } k \geq a} k P(X=k) \\

& \geq \sum_{k \in X(\Omega) \text { s.t. } k \geq a} a P(X=k) \\

&=a \sum_{k \in X(\Omega) \text { s.t. } k \geq a} P(X=k) \\

&=a P(X \geq a)

\end{aligned}

\]

where the first inequality follows from the fact that \(X\) is nonnegative and probabilities are nonnegative, and the second inequality follows from the fact that \(k \geq a\) over the set \(\{k \in X(\Omega)\) s.t. \(k \geq a\}\).

Notation: "s.t." stands for "such that".

Dividing both sides by \(a\), we recover

\[

P(X \geq a) \leq \frac{E X}{a}

\]