Let $U_{n,1},\ldots U_{n,k-1}$ denote the complementary components of $\pi_n(B^c(i,r))$ in $S_n$ that do not contain the infinite end. Call these the finite components. For $n\geq M_r,$ define $\d{S_n(r) =\cup_{j=1}^{k-1} U_{n,j} \cup \pi_{n}(B^c(i,r))}$ Let $p_n\in S_n$ be the projection $\pi_n(i)$ and similarly set $\alpha_0^n=\pi_n(I).$ We have just given distinct names to the point $p$ and the geodesic $\alpha_0.$ Intrinsically, for $n\in {\Bbb{N}}$, $S_n(r)$ is the closed ball of radius $r$ about $p_n$ in $S_n,$ with the finite complementary components added on. It is not yet clear that this will work for $S_{\infty}$, since we have not proved that $S_{\infty}$ is a flute.

The next proposition asserts a kind of strong geometric convergence of the surfaces $S_n$ to $S_{\infty}$.

\[embed\] Given $r>e^{\frac{a}{2}}$ there is a number $N_r\in {\Bbb{N}}$ so that for $m>n\geq N_r$ there exists an isometric bijection $\varphi_{n,m}:S_n(r)\rightarrow S_m(r) $ with $\varphi_{n,m}(\alpha_0^n)=\alpha_0^m$ and $\varphi_{n,m}(p_n)=p_m.$ Furthermore, for $  n\geq N_r$ there exists an isometric embedding $\varphi_n:S_n(r)\rightarrow S_{\infty}$ with $\varphi_{n}(\alpha_0^n)=\alpha_0^{\infty}=\pi_{\infty}(\alpha_0)$ and $\varphi_{n }(p_n)=p_{\infty}=\pi_{\infty}(p).$

A collar of width $2R$ about a simple closed geodesic $\beta$ on a hyperbolic surface $S,$ written $C_S(\beta,R),$ is the set of points on $S$ of distance less than $R$ from $\beta.$ The classical Collar Lemma implies that given $R>0$ 