Existence of group of order $p$ in group of order $pq$, $p>q$

Existence of group of order $p$ in group of order $pq$, $p>q$

This question is related to Question on groups of order $pq$, but is
different.
It references the same exercise, but an earlier part. The exercise is: A
group of order $pq$, $p>q$, contains a subgroup of order $p$ and a
subgroup of order $q$.
The part I'm having trouble with is showing that assuming there is no
subgroup of order $p$ generates a contradiction. It's easy enough to do
with Orbit-Stabilizer Theorem, but that hasn't been introduced in the text
yet, and I haven't been able to find a way to do it with a different
method.
Obviously Cauchy's Theorem trivializes this, but that hasn't been
introduced in the text yet either.
So my question is: How can this problem be solved without Orbit-Stabilizer?
Thanks in advance