I'm looking for a proof of the fact that, if you call $m_n$ the minimal degree of non-trivial irreducible representation of the alternating group $A_n,$ the sequence $\{m_n\}$ tends to $\infty$ for $n \to \infty.$
I have quite basic notions about representation theory (for example, I don't know anything about representations of $S_n$), but I was told that this problem is solvable with elementary notions.
Thanks for your help.