1 $$ Problem 3 [20 marks] Consider the set $\mathcal{5}$ consisting of all bounded and unbounded sequences of complex numbers. 1. Show that $\mathcal{5}$ equipped with the metric ap, q)=\sum_{n=1}^{\infty} \frac{\left|p_{n}-2_{n} ight|H2^{n}\left(1+\left|p_{n}- q_{n} ight ight)} $$ is a metric space. 2. Show that if we replace the factor $1 / 2^{n}$ in $(1)$ with $\mu_{n}>0$ for all $n \in \mathbb{N} $ $$ \sum_{n-1}^{\infty} \mu_{n} $$ is convergent, then $\mathcal(S) $ equipped with $$ \tilde[d}(p, q)=\sum_{n=1}^{\infty} \mu_{n} \frac{\left|p_{n}-Q_{n} ight|}{1+\left|p_{n}- q_{n} ight) $$ is also a metric space. SP.SD. 434 such that the series