A subset $S$, of the real numbers, is bounded when there exists some real number $M$ such that
$|x| \le M$ for all $x \in S$.
(The same for bounded from above and bounded from below separately.)
According to this definition the empty set is bounded. You can chose $M$ whatever you like. The assertion is true, it is vacuously true.
Further more, every real number is an upper bound an every real number is a lower bound, e.g., $26$ is a lower bound and $-45$ an upper bound of the empty set.
To resolve the paradox one needs to note that for $m$ a lower and $M$ and upper bound of some set $S$, one can conclude that $m \le M$ provided that $S$ is non-empty.
One needs some element in $S$ to make the obvious argument work. (For any $s \in S$, we have $m \le s$ and $s \le M$. Thus $m \le M$.)
A takeway is that vacuous truth can have some counterintuitive consequence.