choose $r$ so that ${C_{\theta}}(f)$ is not tangent to the circle $S_r$. (Hence ${M}({C_{\theta}}(f),r)$ is well-defined by Remark \[rmk:tangency\].) Then for $R$ sufficiently large, we have $${M}({C_{\theta}}(g_R),r) = {M}({C_{\theta}}(f),r) \,.$$ Moreover, as $R\rightarrow \infty$, the point labeled $i$ on ${C_{\theta}}(g_R) \cap S_r$ approaches the point labeled $i$ on ${C_{\theta}}(f) \cap S_r$. In particular, if $r$ is sufficiently large, then for $R$ sufficiently large we have $${M}({C_{\theta}}(g_R),r) = {M}(f,\theta)\,.$$

In order to prove Proposition \[prop:disk\], we will need the following (presumably standard) facts from metric topology.

\[lemma:technical\] Let $(X,d)$ be a compact, locally connected metric space. For $x \in
X$ and $\epsilon > 0$, write $B(x,\epsilon)$ for the open ball of radius $\epsilon$ centered at $x$. For any subset $K$ of $X$ and any $\epsilon > 0$, define $$N_{\epsilon}(K) = \bigcup_{x \in K} B(x,\epsilon) \,.$$ For $\beta>0$ and $t \in (0,\beta)$, let $h_t : X\rightarrow {{\mathbb R}}$ be a family of continuous functions converging pointwise (hence uniformly) to some continuous $f : X \rightarrow {{\mathbb R}}$ as $t \rightarrow 0$. Define $$\begin{aligned}
  Z &= \{x\in X {\;:~}f(x)=0\},\\
  Y_t &= \{x\in X {\;:~}h_t(x)=0\}.
  \end{aligned}$$ Then:

1.  For all $\epsilon > 0$, if $t$ is sufficiently close to $0$ then $Y_t \subset
            N_\epsilon(Z)$.

