TITLE: Why do all compact connected surfaces have a triangulation?
QUESTION [23 upvotes]: Does anyone know a reference for a relatively elementary proof that every compact connected surface can be triangulated? By "elementary," I mean that I could present at least a sketch to undergraduates taking a first semester topology course. 
When one proves the classification of surfaces, invariably one assumes that the surface is triangulable or smooth, but I'd like to justify this missing step.

REPLY [12 votes]: This paper by Carsten Thomassen (snatched from the MO thread I mentioned in the comments) may contain the most elementary proof.