\begin{frame} \frametitle{Infinite Terms as Metric Space} \begin{definition} We define a \alert{metric} $\alert{d}$ on $\iter$ by: \begin{align*} d(s,t) = 2^{-|p|} \text{ where $p$ is the highest position such that $s(p) \ne t(p)$} \end{align*} \end{definition} \pause Note that $d(s,t) = 0 \;\Longleftrightarrow\; s = t$. \pause \begin{example} \begin{center} \begin{tikzpicture}[level distance=6mm,inner sep=0.5mm,sibling distance=10mm] \node (s) {$f$} child { node {$c$} child { node {$c$} child { node {$c$} child { node {$c$} child { node {$\vdots$} } } } } } child { node {$c$} child { node (b) {$b$} } }; \node [left of=s,node distance=7mm] {$s = $}; \node (t) [right of=s,node distance=30mm]{$f$} child { node {$c$} child { node {$c$} child { node {$c$} child { node {$c$} child { node {$\vdots$} } } } } } child { node {$c$} child { node (c) {$c$} child { node {$b$} } } }; \node [left of=t,node distance=7mm] {$t = $}; \end{tikzpicture} \end{center} \vspace{-2ex} \pause The first difference is at depth $2$, hence $d(s,t) = 2^{-2} = 0.25$. \end{example} \end{frame}