Step 1

A) Let \(\displaystyle{\left({X},\ {d}\right)}\) be a metric space A be a subset of x.

We define diameter of A to be

Sup \(\displaystyle{\left\lbrace{d},\ {\left({x},\ {y}\right)}{\mid}{x},\ {y}\in{A}\right\rbrace}\)

Let \(\displaystyle\cup\) be an open boll with center at \(\displaystyle{x}_{{{0}}}\) and radious \(\displaystyle{r}{>}{0}\)

Then, diameter of \(\displaystyle\cup={2}{r}\)

We choose x, y to be diametrically opposite two points whose midpoint is \(\displaystyle{x}_{{{0}}}\)

Then \(\displaystyle{d}{\left({x},\ {y}\right)}\) atfains 2r.

Also, for any \(\displaystyle{x},\ {y}\in\cup\)

\(\displaystyle{d}{\left({x},\ {y}\right)}\le{d}{\left({x},\ {x}_{{{0}}}\right)}+{d}{\left({x}_{{{0}}},\ {y}\right)}\)

\(\displaystyle\le{r}+{r}={2}{r}\)

\(\displaystyle\therefore\ {d}{i}{a}{m}{\left(\cup\right)}={2}{r}\)

Step 2

B) Let \(\displaystyle{\left({x}_{{{n}}}\right)}_{{{n}\not\in{{{N}}}}}\) be a squence which is convergent in \(\displaystyle{\left({x},\ {d}\right)}\)

We assume that, it converges to \(\displaystyle{x}\in x\)

Choose \(\displaystyle\sum={1}\)

Then \(\displaystyle\exists{N}\not\in{{{N}}}\)

s.t \(\displaystyle\forall{h}\geq\ {d}{\left({x}_{{{n}}},\ {x}\right)}{ < }{1}\)

We take

\[N1=Sup\begin{cases}d(x,\ x_{1}), & d(x,\ x_{2})\\ \dots, & d(x,\ x_{n-1}),1\} \end{cases}\]

Then for any \(\displaystyle{x}_{{{n}}}\) in the sep

\(\displaystyle{d}{\left({x}_{{{n}}},\ {x}\right)}{ < }{1}\)

So, \(\displaystyle{\left({x}_{{{n}}}\right)}_{{{n}\not\in{{{N}}}}}\) is bounded