(a) Let $\left(x_{n}\right)$ be a bounded sequence of real numbers. Show that $\left(x_{n}\right)$ has a convergent subsequence.

(b) Let $\left(z_{n}\right)$ be a bounded sequence of complex numbers. For each $n \geqslant 1$, write $z_{n}=x_{n}+i y_{n}$. Show that $\left(z_{n}\right)$ has a subsequence $\left(z_{n_{j}}\right)$ such that $\left(x_{n_{j}}\right)$ converges. Hence, or otherwise, show that $\left(z_{n}\right)$ has a convergent subsequence.

(c) Write $\mathbb{N}=\{1,2,3, \ldots\}$ for the set of positive integers. Let $M$ be a positive real number, and for each $i \in \mathbb{N}$, let $X^{(i)}=\left(x_{1}^{(i)}, x_{2}^{(i)}, x_{3}^{(i)}, \ldots\right)$ be a sequence of real numbers with $\left|x_{j}^{(i)}\right| \leqslant M$ for all $i, j \in \mathbb{N}$. By induction on $i$ or otherwise, show that there exist sequences $N^{(i)}=\left(n_{1}^{(i)}, n_{2}^{(i)}, n_{3}^{(i)}, \ldots\right)$ of positive integers with the following properties:

• for all $i \in \mathbb{N}$, the sequence $N^{(i)}$ is strictly increasing;

• for all $i \in \mathbb{N}, N^{(i+1)}$ is a subsequence of $N^{(i)} ;$ and

• for all $k \in \mathbb{N}$ and all $i \in \mathbb{N}$ with $1 \leqslant i \leqslant k$, the sequence

$$\left(x_{n_{1}^{(k)}}^{(i)}, x_{n_{2}^{(k)}}^{(i)}, x_{n_{3}^{(k)}}^{(i)}, \ldots\right)$$

converges.

Hence, or otherwise, show that there exists a strictly increasing sequence $\left(m_{j}\right)$ of positive integers such that for all $i \in \mathbb{N}$ the sequence $\left(x_{m_{1}}^{(i)}, x_{m_{2}}^{(i)}, x_{m_{3}}^{(i)}, \ldots\right)$ converges.