= Solution
The <events> $\{X=k\}$, for $k\in\mathbb Z$, are pairwise disjoint. Their union is $\{X\in\mathbb Z\}$, and the <integers> form a <countable set>. By <countable additivity>,
$$
\boxed{\mathbb P(X\in\mathbb Z)=\sum_{k\in\mathbb Z}\mathbb P(X=k)=0.}
$$
Countability is the necessary justification: one cannot apply this argument to an arbitrary uncountable union of zero-probability singletons. This is a special case of <countable sets have zero probability under a density>.
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