How would I prove something like this:
$\forall x \in \mathbb{R}, \exists n \in \mathbb{Z}$ such that $|x − n| < \frac{1}{4}$
I read it as "For all real numbers $x$, there exists an integer $n$, such that the absolute value of $x-n$ is less than $\frac{1}{4}$."
Is this correct?
I understand this is such an easy proof and I assume it is obviously true, but I don't really understand where to start it or how to prove it. Should I do it by cases? Should I do it by contradiction?
You can't! That statement is wrong. If $x=1.5$, there are no integers that are within a quarter of $x$!