(a) Define what it means for a metric space $(X, d)$ to be complete. Give a metric $d$ on the interval $I=(0,1]$ such that $(I, d)$ is complete and such that a subset of $I$ is open with respect to $d$ if and only if it is open with respect to the Euclidean metric on $I$. Be sure to prove that $d$ has the required properties.

(b) Let $(X, d)$ be a complete metric space.

(i) If $Y \subset X$, show that $Y$ taken with the subspace metric is complete if and only if $Y$ is closed in $X$.

(ii) Let $f: X \rightarrow X$ and suppose that there is a number $\lambda \in(0,1)$ such that $d(f(x), f(y)) \leqslant \lambda d(x, y)$ for every $x, y \in X$. Show that there is a unique point $x_{0} \in X$ such that $f\left(x_{0}\right)=x_{0}$.

Deduce that if $\left(a_{n}\right)$ is a sequence of points in $X$ converging to a point $a \neq x_{0}$, then there are integers $\ell$ and $m \geqslant \ell$ such that $f\left(a_{m}\right) \neq a_{n}$ for every $n \geqslant \ell$.