global-analysis-and-the-banach-manifold-of-class-h1-curvers

diff --git a/sections/structure.tex b/sections/structure.tex
@@ -59,11 +59,12 @@ approximating the curve
   \gamma : I & \to     \RR^2      \\
            t & \mapsto (t, 1 - t)
 \end{align*}
-with ``step curves'' \(\gamma_n : I \to \RR^n\) for larger and larger values of
-\(n\), as shown in figure~\ref{fig:step-curves}.
+with ``staircase curves'' \(\gamma_n : I \to \RR^n\) for larger and larger
+values of \(n\), as shown in figure~\ref{fig:step-curves}: clearly \(\gamma_n
+\to \gamma\) in the uniform topology, but \(\ell(\gamma_n) = 2\) does not
+approach \(\ell(\gamma) = \sqrt 2\) as \(n\) approachs \(\infty\).
 
-% TODO: Add a figure and a caption explaining why length is discontinuous
-\begin{figure}\label{fig:step-curves}
+\begin{figure}[h]\label{fig:step-curves}
   \centering
   \begin{tikzpicture}
     \draw (4, 1) -- (1, 4);
@@ -81,7 +82,8 @@ with ``step curves'' \(\gamma_n : I \to \RR^n\) for larger and larger values of
     \node[above] at (1.5, 4.3) {$\sfrac{1}{n}$};
   \end{tikzpicture}
   \caption{A diagonal line representing the curve \(\gamma\) overlaps a
-  staircase-like curve \(\gamma_n\), whose steps measure \(\sfrac{1}{n}\).}
+  staircase-like curve \(\gamma_n\), whose steps measure \(\sfrac{1}{n}\) in
+  width and height.}
 \end{figure}
 
 The issue with this particular example is that while \(\gamma_n \to \gamma\)