 Solution

A simplicial map $g:K\to L$ is a simplicial approximation to $f:|K|\to |L|$ whenfor every vertex $v$, where $\mathrm{st}$ denotes the open star in a simplicial complex.

Fix $x$ in a simplex with vertices $v_0,\dots ,v_q$, retaining only vertices with positive barycentric coordinate at $x$. Then $x\in {\mathrm{st}}_K(v_i)$ for every $i$, so $f(x)$ lies in every ${\mathrm{st}}_L(g(v_i))$. The vertices $g(v_i)$ and the vertices carrying the barycentric coordinates of $f(x)$ consequently span a common simplex of $L$. Both $f(x)$ and $|g|(x)$ lie in its convex realization. The straight-line homotopy from a simplicial approximationis therefore well-defined and continuous, with endpoints $f$ and $|g|$.