Since $E \subseteq cl(E) $, then it is immediate that diam $(E) \leq $ diam(cl($E))$.
I only need to show that assuming diam $(E) < $ diam(cl($E))$ will lead to contradiction then I can conclude that diam $(E)= $ diam(cl($E))$.
So suppose that diam(cl($E$)) > diam($E$). Then there exist a $p,q \in $ cl($E$) such that $ d(p,q) > $diam($E$). By def of cl($E$), there exist a sequence of $\{p_n\}, \{ q_n \} \in E$ such that $ p_n , q_n \to p, q ~~ $ respectively as $ n \to \infty $.
Also note that by triangle inequality ,we have
$$ (1) ~~~~~~ d(p,q) – [d(p,p_n) + d(q,q_n)] \leq d(p_n,q_n) \text{ for all } n.$$
Since $ p_n, q_n \to p,q $ , we have $ d(p,p_n) + d(q,q_n) \to 0 $ as $ n \to \infty$.
Since $ d(p,q) > $ diam(cl($E$)), we can choose some $p_n, q_n $ such that
$$ d(p,q) – [d(p,p_n) + d(q,q_n)] > \text{diam}(E).$$
Then by (1) above, we have
$$ d(p_n,q_n) > \text{diam}(E),$$
a contradiction. Thus the result holds.
Is my proof correct? and is there a shorter way to do this?? thank you very much.
Best Answer
Suppose that $x,y\in\operatorname{cl}(E)$ and fix $\varepsilon>0$. Then, there exist $a,b\in E$ such that \begin{align*} d(a,x)<&\,\frac{\varepsilon}{2},\\ d(b,y)<&\,\frac{\varepsilon}{2}. \end{align*} The triangle inequality then implies that $$d(x,y)\leq d(x,a)+d(a,b)+d(b,y)\leq d(x,a)+\underbrace{\sup_{\hat a,\hat b\in E}d(\hat a,\hat b)}_{=\operatorname{diam}(E)}+d(b,y)\leq\operatorname{diam}(E)+\varepsilon.$$ Taking supremum over $x,y\in\operatorname{cl}(E)$, one has that $$\operatorname{diam}(\operatorname{cl}(E))=\sup_{x,y\in\operatorname{cl} E}d(x,y)\leq\operatorname{diam}(E)+\varepsilon.$$ Since $\varepsilon>0$ is arbitrary, it follows that $$\operatorname{diam}(\operatorname{cl}(E))\leq\operatorname{diam}(E).$$