Proof

If $a \neq 0$ is an integer and $b$ its divisor $b\mid a$, then, by definition, there exists an integer $q\neq 0$ with \[a=qb.\]
Since $q\ge 1$, it follows from the properties of the absolute value and the rules of calculation with inequalities that $|a|=|q||b|\ge |b|.$
Since $|a|$ is finite and since $|b| > 0$, there are only finitely many $b$ such that $|a|\ge |b|$.
Thus, the number of divisors of $|a|$ is also finite.
∎

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Landau, Edmund: "Vorlesungen über Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927