Let $K$ and $L$ be simplicial complexes, $m=\dim K$, and $h:|K|\rightarrow |L|$ be a continuous map. Show that $h$ is homotopic to a map carrying $K$ into $L^{(m)}$, the $m$-skeleton of $L$.
I'm having a lot of trouble with this problem. I can see why it is true but I'm getting stuck while writing the proof on the case where $h$ doesn't satisfy the star condition (that for each $u\in\operatorname{vert}(K)$ there is a $v\in\operatorname{vert}(K)$ with $h(\operatorname{st}(u))\subseteq \operatorname{st}(v)$).
For example, suppose we are mapping a filled in triangle $abc$ into two filled in triangles, $x_1y_1z_1$ and $x_2y_2z_2$, joined by an edge, $x_1x_2$. Assume further that $h(a)\in\operatorname{int}(x_1y_1z_1)$ and $\{h(b),h(c)\}\subseteq \operatorname{int}(x_2y_2z_2)$. I know what we need to do is "pull $h(a)$ into $x_2y_2z_2$ through the edge $x_1x_2$", then map $h(a)h(b)h(c)$ to $x_2y_2z_2$. But I don't know how to word that mathematically, and certainly can't think of how to generalize it. Maybe I'm taking the wrong approach.
The barycentric subdivision of a simplicial complex $K$ is constructed by adding a vertex to the centroid of every simplex and the cone to its boundary. Iterating this operation $n$ times gives the $n^\text{th}$ barycentric subdivision $\operatorname{Sd}^n(K)$.
The simplicial approximation theorem states that if $g:|K|\rightarrow |L|$ is continuous, where $K$ and $L$ are simplicial complexes, then there is an integer $n$such that $g$ has a simplicial approximation $f:\operatorname{Sd}^n(K)\rightarrow L$.
By this theorem, we can take a simplicial approximation $f:\operatorname{Sd}^n(K)\rightarrow L$ of $h$ for some appropriately large $n$. Barycentric subdivision does not affect the dimension of a space, so $f$ maps $\operatorname{Sd}^n(K)$ into $L^{(m)}$, so in particular, $f$ maps $K$ into $L^{(m)}$. We can prove without much trouble that a continuous map $g$ is homotopic to any simplicial approximation of $g$, so it suffices to apply this to $h$ and $f$.