Prove that two paths on opposing corners of the unit square must cross. I'm looking for a simple argument to the following:
Given two (continuous) paths on the unit square, one from (0,0) to (1,1) and the other from (1,0) to (0,1), prove that the paths cross at some point $(x_0, y_0)$.
I have constructed a topological argument for why this is true using compactness (and a proof by contradiction), but the person I'm working with seems to think there is a "simple" and "well-known" argument that says the two paths must cross.  I haven't been able to find such a result.  Does anyone know of one?
Thanks!

The result is Lemma 2 of this paper by Maehara, which uses the Brouwer Fixed Point Theorem.  (I'm not sure if this argument qualifies as "simple" since this theorem is usually proved using homology.)  See also this MO thread.
