This question feels completely trivial and I am somewhat embarrassed to be asking it, but I am having a brain dead moment and failing to prove what I'm sure is a completely trivial statement about groups; I think the proof will probably be a 1- or 2-line affair.
I have a finite connected Cayley graph with $n$ vertices (or elements), with generating set $S \cup S^{-1}$ (so we assume $S$ spans the group), and I want to write a general element $x$ in terms of the generators. I wish to show that we can always write as a product of generators $x = g_1 \ldots g_m$, where $m \leq n/2$; i.e. we can always write an element as a product of generators of length at most half the order of the group.
For the theorem I am using this to prove, I also assume $S \cup S^{-1}$ is closed under conjugation, i.e. contains full conjugacy classes, but I don't think this is needed here. In case I'm actually not being stupid (though I'm 95% sure I am), the shorter the proof possible, the better.
Thanks for your help, and for putting me out of my misery!