Exercise 3.2.5

Let $F\subseteq R$ be closed and suppose $(x_n)$ is a Cauchy sequence in $F$, since Cauchy sequences converge $(x_n)\to x$ and finally since $x\in F$ since $F$ contains its limit points.

Now suppose every Cauchy sequence $(x_n)$ in $F$ converges to a limit in $F$ and let $l$ be a limit point of $F$, as $l$ is a limit point of $F$ there exists a sequence $(y_n)$ in $F$ with $\lim <!--\; FUNCTION\; APPLICATION\; -->(y_n)=l$. since $(y_n)$ converges it must be Cauchy, and since every Cauchy sequence converges to a limit inside $F$ we have $l\in F$.