It's well known that every subgroup $G$ of $S_n$ has a generating set of size at most $n-1$ and that this generating set can be found algorithmically (by Jerrum's filter)

I have heard many times a stronger result, which is that for all $n>3$, every subgroup $G$ of $S_n$ has a generating set of size at most $n/2$. Note that this would be tight: for example, $n/2$ disjoint transpositions cannot be minimized further.

However, as far as I can tell, none of the places I have seen this theorem give a proof, nor do their references. Can anyone point me to a proper proof, or give one, for this seemingly important theorem?