Uniform Extensions into Complete Metric Spaces (7.43.2)

James Munkres Topology, chapter 7.43, exercise 2:

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, with $Y$ complete. Show that if $A⊆X$ and $f : A→Y$ is uniformly continuous, there exists a uniformly continuous extension of $f$ to $\overline{A}$.

Proof: Let $(x_n)$ be a Cauchy sequence in $X$; we show $(f(x_n))$ is a Cauchy sequence in $Y$. Let $ε > 0$. Then if $δ > 0$ is such that $d_Y(f(a),f(b)) < ε$ for all $a,b∈A$ such that $d_X(a,b) < δ$, and $N$ is such that $d_X(x_n,x_m) < δ$ for all $n,m≥N$, we see $d_Y(f(x_n),f(x_m)) < ε$ for all $n,m≥N$, so that $f(x_n)$ is Cauchy.

For all $x∈\overline{A}$, choose some Cauchy sequence $(x_n)$ in $A$ converging to $x$. Then $f(x_n)→y_x$ since $Y$ is complete. Define $g: \overline{A}→Y$ by $g(x)=y_x$. Since $f$ is continuous, $y_x=f(x)$ for all $x∈A$. Now it suffices to show $g$ is uniformly continuous. Let $ε > 0$; let $δ > 0$ be such that $d_Y(f(a),f(b)) < ε/3$ whenever $a,b∈A$ are such that $d_X(a,b) < δ$. Let $x,y∈\overline{A}$ be such that $d_X(x,y) < δ/3$; let $(x_n)→x$ and $(y_n)→y$ be the chosen sequences as before. Choose $n$ such that $$d_Y(g(x),g(x_n)),d_Y(g(y),g(y_n)) < \text{min }\{ε/3,δ/3\}$$ We see $d_X(g(x_n),g(y_n)) = d_X(f(x_n),f(y_n)) < ε/3$ since $d_X(x_n,y_n) ≤ d_X(x_n,x)+d_X(x,y)+d_X(y,y_n)$. Finally, we observe $$d_Y(g(x),g(y)) ≤ d_Y(g(x),g(x_n)) + d_Y(g(x_n),g(y_n)) + d_Y(g(y_n),g(y)) < ε$$