Sources that representations on $S_n$ are real-valued

Question

I've been told that representations of $S_n$ are real-valued and that there are some sources for that. However, I can't find anything, does anybody have a book / paper that proves this?

Thanks in advance.

Answer 1

Every element is conjugate to its inverse in $S_n$. Thus $\chi(x^{-1})=\chi(x)$ for all $x\in S_n$ and characters $\chi$. But $\chi(x^{-1})=\overline{\chi(x)}$, and we are done.