\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}