find the supremum and infimum of a set S

Question

The question demands computing the supremum and infimum of the set $S=\{x:x^2\leq7, x \text{ is rational}\}$. My approach is to estimate the set as an interval say $[-\sqrt7,\sqrt7]$ but the rationality part is confusing, I of course considered that the interval is dense of rationals. Some help on how to get the supremum and infimum. Should I just estimate a number say $-\sqrt 7$ but then it's irrational.

Answer 1

$\sqrt 7$ is an upper boud for $S$, because $x\in S$ implies $x^2\le 7$, implies $x\le\sqrt 7$. Any number $a<\sqrt 7$ is not an upper bound for $S$, because you can show (how!?) that there exist rational numbers $x$ with $x>a$ and $x^2\le 7$. Therefore, $\sup S=\sqrt 7$. Similar for the infimum.