Rotman's exercise 2.8 " $S_n$ cannot be imbedded in $A_{n+1}$"

Rotman's exercise 2.8 " $S_n$ cannot be imbedded in $A_{n+1}$"

This question is about the (in)famous Rotman's exercise 2.8 in "An
Introduction to the Theory of Groups" I've search and found similar
questions here and in MO, but none of them contains a valid proof. (Does
$S_n$ belong as a subgroup to $A_{2n+1}$?)
According to Rotman, a valid proof can only use the concepts introduced up
to this exercise: cycle permutations, factorization of permutations, odd
and even permutations, semigroups, groups, homomorphism and subgroups.
Cosets, Lagrange's theorem, normal subgroups, and so on are not yet
introduced. I stress this point because all of the proofs I've seen use
Lagrange or actions, on cosets.
Now my attempt is to use exercise 2.7 (solved) which is about a proof that
$A_n$ ($n>2$) is generated by all the 3-cycle and exercise 2.4 (solved) "
if $S$ is a proper subgroup of G then $\langle G \setminus S\rangle=G$ "
in this way:
Suppose that for every $\phi : S_n \to A_{n+1}$ imbeddings, all the $
3$-cycles are contained in $\operatorname{Im}\phi $, then the assertion is
proved by absurd. But I can't find a way to prove if it is possible or
either find a counterexample to this kind of approach.
If someone has another proof which use only basics concepts is well
accepted of course, but I mainly need some hints about correctness or not
of my reasoning and how to proceed if it is correct. Thank you in advance.