(a) What does it mean to say that the sequence $\left(x_{n}\right)$ of real numbers converges to $\ell \in \mathbb{R} ?$

Suppose that $\left(y_{n}^{(1)}\right),\left(y_{n}^{(2)}\right), \ldots,\left(y_{n}^{(k)}\right)$ are sequences of real numbers converging to the same limit $\ell$. Let $\left(x_{n}\right)$ be a sequence such that for every $n$,

$$x_{n} \in\left\{y_{n}^{(1)}, y_{n}^{(2)}, \ldots, y_{n}^{(k)}\right\}$$

Show that $\left(x_{n}\right)$ also converges to $\ell$.

Find a collection of sequences $\left(y_{n}^{(j)}\right), j=1,2, \ldots$ such that for every $j,\left(y_{n}^{(j)}\right) \rightarrow \ell$ but the sequence $\left(x_{n}\right)$ defined by $x_{n}=y_{n}^{(n)}$ diverges.

(b) Let $a, b$ be real numbers with $0<a<b$. Sequences $\left(a_{n}\right),\left(b_{n}\right)$ are defined by $a_{1}=a, b_{1}=b$ and

$$\text { for all } n \geqslant 1, \quad a_{n+1}=\sqrt{a_{n} b_{n}}, \quad b_{n+1}=\frac{a_{n}+b_{n}}{2} \text {. }$$

Show that $\left(a_{n}\right)$ and $\left(b_{n}\right)$ converge to the same limit.