Exercise 4.11

Let $𝜀>0$ and let $\delta >0$ such that $d_Y\left(f(x),f(y)\right)<𝜀$ if $d_X(x,y)<\delta$. Let $\{x_n\}$ be a Cauchy sequence in $X$. Then there is an integer $N$ such that $d_X(x_n,x_m)<\delta$ if $n\ge N$ and $m\ge N$. Hence $d_Y\left(f(x_n),f(x_m)\right)<𝜀$ if $n\ge N$ and $m\ge N$, so $\{f(x_n)\}$ is a Cauchy sequence in $Y$.

Let $E$ be a dense subset of a metric space $X$, and let $f$ be a uniformly continuous mapping of $X$ into a complete metric space $Y$. We can extend $f$ to a function on all of $X$ as follows. Let $x\in X$ and $\{p_n\}$ be a sequence in $E$ which converges to $x$. Then $\{f(p_n)\}$ is a Cauchy sequence in $Y$ which converges to $y\in Y$ since $Y$ is complete. This sets up a well-defined function $g$ from $X$ to $Y$ since if $\{p_n^{\prime }\}$ is another sequence in $X$ converging to $x$, then $\{f(p_n)\}$ converges to $y^{\prime }\in Y$. Since $f$ is uniformly continuous on $D$, for $𝜀>0$ there is $\delta >0$ such that if $p,p^{\prime }\in E$ with $d_X(p,p^{\prime })<\delta$ then $d_Y\left(f(p),f(p^{\prime })\right)<𝜖∕3$. There is an integer $N$ such that $d_X(p_n,x)<\delta ∕2$ and $d_X(p_n^{\prime },x)<\delta ∕2$ for all $n\ge N$. Then for such $n$ we have

Also, there is an integer $M\ge N$ such that for all $n>M$ we have $d_Y\left(y,f(p_n)\right)<𝜀∕3$ and $d_Y\left(y^{\prime },f(p_n^{\prime })\right)<𝜀∕3$. Hence, for large enough $n$ we have

Considering the constant sequence $\{x\}$ for $x\in E$, we see that $g$ extends $f$ to all of $X$. To see that $g$ is uniformly continuous, let $𝜀>0$. Since $f$ is uniformly continuous on $E$, there is $\delta >0$ such that if $p,p^{\prime }\in E$ with $d_X(p,p^{\prime })<\delta$ then $d_Y\left(f(p),f(p^{\prime })\right)<𝜖∕3$. Let $x,x^{\prime }\in X$ such that $d_X(x,x^{\prime })<\delta ∕3$, and let $\{p_n\}$ and $\{p_n^{\prime }\}$ be sequences in $E$ which converge to $x$ and $x^{\prime }$, respectively. Let $n$ be a large enough integer so that $d_X(p_n,x)<\delta ∕3$ and $d_X(p_n^{\prime },x^{\prime })<\delta ∕3$. Then

so that $d_Y\left(f(p_n),f(p_n^{\prime })\right)<𝜖∕3$. Also, if $n$ is large enough we have $d_Y\left(g(x),f(p_n)\right)<𝜀∕3$ and $d_Y\left(g(x^{\prime }),f(p_n^{\prime })\right)<𝜀∕3$. Hence, for large enough $n$ we have

Hence $g$ is uniformly continuous on $X$.