Solution
= Solution
The <mesh of a simplicial complex> is the largest diameter of one of its realized simplices, or the supremum when $K$ is not finite:
$$
\mu(K)=\sup_{\sigma\in K}\operatorname{diam}|\sigma|.
$$
If $\dim K=n$, then
$$
\mu(K')\leq\frac{n}{n+1}\mu(K).
$$
Iterating gives
$$
\boxed{\mu(K^{(r)})\leq
\left(\frac{n}{n+1}\right)^r\mu(K)\longrightarrow0.}
$$