Prove the following without Lagrange's theorem:
If $H$ is a subgroup of odd order of $S_n$, then it is a subgroup of $A_n$.
So this proof is pretty trivial if you have Lagrange's theorem, but we haven't covered that or cosets yet (our course is going super slow), so I probably shouldn't use it. I am having difficulty just deleting knowledge of the theorem from my brain, and I can't just reconstruct the relevant parts of Lagrange's theorem because it needs cosets. How might I prove this?
For reference, this is question 5.24 in Gallian's Contemporary Abstract Algebra 8th ed.