exercise 1.4b

inputs/tutorial_02.tex

@@ -229,4 +229,28 @@ Clearly $d_u$ is a metric.
 \begin{claim}
     There exists a countable dense subset.
 \end{claim}
-\todo{handwritten solution}
+\begin{subproof}
+    Fix a metric $d_X$ on $X$ defining its topology.
+    Let
+    \[
+    C_{m,n} \coloneqq  \{f \in  \cC(X,Y) : \forall x,y \in X.~\left( d_X(x,y) < \frac{1}{m+1} \implies d(f(x), f(y)) <\frac{1}{n+1}\right) \}.
+    \]
+
+    Choose $X_m \subseteq X$ finite with $X \subseteq \bigcup_{x \in X_m} B_{\frac{1}{m+1}}(x)$.
+    Let $D_{m,n} \subseteq C_{m,n}$ be countable,
+    such that for every $f \in C_{m,n}$ and every $\eta >  0$,
+    there is $g \in D_{m,n}$ with $d(f(y), g(y)) < \frac{\eta}{3}$
+    for each $y \in X_m$.
+    Then $\bigcup_{m,n} D_{m,n}$ is dense in $\cC(X,Y)$:
+    Indeed if $f \in \cC(X,Y)$ and $\eta > 0$,
+    we finde $n > \frac{3}{\eta}$ and $m$ such that $f \in C_{m,n}$,
+    since $f$ is uniformly continuous.
+    Let $g \in D_{m,n}$ be such that $\forall y \in X_m.~d(f(y), g(y)) < \frac{1}{n+1}$.
+    We have $d_u(f,g) \le \eta$,
+    since for every $x \in X$, we find $y \in X_m$ with $d_X(x,y) < \frac{1}{m+1}$,
+    hence
+    \begin{IEEEeqnarray*}{rCl}
+        d_Y(f(x), g(x)) &\le& d_Y(f(x), f(y)) + d_Y(f(y), g(y)) + d_Y(g(y), g(x))\\
+                        &\le& \frac{1}{n+1}  + \frac{1}{n+1} + \frac{1}{n+1} \le \eta.
+    \end{IEEEeqnarray*}
+\end{subproof}