1 $$ Problem 3 E20 marks] Consider the set $\mathcal{5}$ consisting of all bounded and unbounded sequences of complex numbers. 1. Show that $\mathcal{S}$ equipped with the metric p. 9-\sum_(n=1}^{\infty] \frac{\left|p_{n}-Q_(n} ight|}2^{n}\left(+\left|p_(n)- q_{n} ight ight) $$ 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} $ $$ \sum _{n-1}*(\infty} \mu_{n} is convergent, then $\mathcal{S$ equipped with $$ tiided (p, q)=\sum_(n=1}^{\infty) \m_{n} \frac{\left|p_{n}-q_{n} ight|H1+\left|p_{n} - 9_{n} ight| $$ is also a metric space. SP.SD.434 such that the series $$