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> Prove there exists infinitely many real numbers $a$ such that $a(a - 3\{a\})$ is an integer. $\{a\}$ = fractional part of a number.
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Here is my proof. Are there any fallacies that I’m missing? Alternate proofs are also welcome :)

Let $S$ be the set of reals obeying this property.
Let $\lambda$ be the greatest real in this set.

Then let $ \alpha = \lambda + (1 - \{\lambda\})$

(In essence, $\alpha$ takes the number and rounds up to the nearest integer.)

Since $\alpha > \lambda$, $\alpha$ is not in S.

But $\alpha$ is an integer and all integers obey this property.

Thus, $\alpha$ is a member of $S$,
which completes the proof by contradiction.