No cycles in finite coxeter graphs

Question

Is there an elementary (no consideration of root systems involved) proof of the fact that the graph of an finite coxeter system doesn't entail any cycle? I got as far as this: If there were any cycle $s_1,\dots,s_p$, consider the element $s_1\dots s_p$. It's probably going to be of infinite order (but I can't say why) and therefore the coxeter group isn't finite. Alternatively, $(s_1\dots s_p)^r$ could yield to elements of increasing length contradicting the fact that finite coxeter groups entail an element of maximal length.