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Prove or disprove:

For any real number $x$, there exist integers $a$ and $b$ such that $ae + b\pi=x$.

It certainly seems improbable, but how does one prove it?

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1 Answer 1

If every $x \in \mathbb{R}$ could be written as $ae+b\pi$ for some $a,b \in \mathbb{Z}$, then we would have a surjection from $\mathbb{Z}^2$ (which is countable) to $\mathbb{R}$ (which is uncountable).

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