Exercise 1.4.7

Proof. If $\left(Y,{d|}_{Y\times Y}\right)$ is complete then every Cauchy sequence in $Y$ also converges in $Y$. Given any convergent sequence of elements of $Y$ that converges to say $x$, we know that it is also Cauchy in $Y$ (Lemma 1.4.7) and hence convergent in $Y$, that is, $x\in Y$. Thus $Y$ is closed in $X$ (by Proposition 1.2.15(b)). □

Proof. Let $\left(y_n\right)$ be a Cauchy sequence of elements in $Y$. Since $X$ is complete we know that $\left(y_n\right)$ converges to some $x\in X$. From Proposition 1.2.10 we can then conclude that $x\in \mathrm{Int}<!--\; FUNCTION\; APPLICATION\; -->(Y)$ or $x\in \mathit{\partial Y}$. If $x\in \mathrm{Int}<!--\; FUNCTION\; APPLICATION\; -->(Y)$ then by Lemma 1 (b) we know $x\in Y$. If $x\in \mathit{\partial Y}$ then we know that $x\in Y$ as $Y$ is closed. In both cases we get $x\in Y$. Thus every Cauchy sequence in $Y$ also converges in $Y$, and hence $Y$ is complete. □