TITLE: Groups and generating sets
QUESTION [18 upvotes]: 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!

REPLY [2 votes]: I can think of another proof in your specific situation ($S\cup S^{-1}$ closed under conjugation), but I don't see how to modify it to not need that assumption, and I prefer ccc's proof.
For $g\in G$, let $|g|_S$ denote the minimum length of a word in elements of $S\cup S^{-1}$ representing $g$.  Note that $S\cup S^{-1}$ being closed under conjugation implies that for any $g,h\in G$ we have $|g^{-1}hg|_S = |h|_S$.
Suppose $x\in G$ with $|x|_S = m>\frac{n}{2}$ and write $x = g_1 g_2\ldots g_m$ with $g_i\in S\cup S^{-1}$.  Then the prefixes $g_1\ldots g_i$ are all distinct.  Also $g_m^{-1}\ldots g_1^{-1}$ is a shortest word representing $x^{-1}$ and so the prefixes $g_m^{-1}\ldots g_i^{-1}$ are all distinct.  
Now, including $1_G$, we have found $2m+1\geq n+2$ elements of $G$, and so $g_1\ldots g_i =_G g_m^{-1}\ldots g_j^{-1}$ for some $1\leq i,j <m$.  But then $x = g_1 \ldots g_i g_{i+1}\ldots g_m = g_m^{-1}\ldots g_j^{-1} g_{i+1}\ldots g_m$.  So $x = g_m^{-1}hg_m$, where $|h|_S \leq m - 2$.  But this implies $|x|_S \leq m-2$, a contradiction.