how to approximate continuous maps with simplicial maps

Most commonly studied Topological spaces admit the structure of a simplicial complex. If K and L are Simplicial complexes, and f: |K| → |L| is some continuous map between their topological realization, it will be very useful to be able to homotope f to a Simplicial map. In general, this is not always possible. But it is if we pass to a ‘sufficiently fine’ subdivision of K:

Simplicial approximation theorem. Let K and L be simplicial complexes, where K is finite, and let f: |K| → |L| be a continuous map. Then there is some subdivision K' of K and a simplicial map g: K'→ L such that |g| is homotopic to f.