= Solution
The <Simplicial approximation theorem> states that if $K$ is a finite simplicial complex, $L$ is a simplicial complex, and $f:|K|\to|L|$ is continuous, then for some $r\geq0$ there is a simplicial approximation $g:K^{(r)}\to L$ to $f$.
The target open stars form an open cover of $|L|$, so their inverse images under $f$ form an open cover of the compact metric space $|K|$. The <Lebesgue number lemma> supplies $\delta>0$ such that every subset of diameter below $\delta$ lies in one inverse image. Choose $r$ so large that every vertex star in $K^{(r)}$ has diameter below $\delta$; this is possible because its diameter is at most twice $\mu(K^{(r)})$ and the mesh tends to zero.
For each vertex $v$ choose a vertex $w(v)$ of $L$ such that
$$
f(\operatorname{st}_{K^{(r)}}(v))
\subseteq\operatorname{st}_L(w(v)).
$$
If $v_0,\ldots,v_q$ span a simplex, a point in its relative interior belongs to all their open stars. Its image belongs to all the open stars of $w(v_0),\ldots,w(v_q)$, so these target vertices span a simplex. Thus $v\mapsto w(v)$ extends to a simplicial map $g$, and its defining inclusion says exactly that $g$ approximates $f$. This proves the theorem.
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