I could not prove the following question could you please help me?
Best Regards
Let $X, d(x, y)$ be a metric space. By definition, diameter of a bounded set $A ⊂ X$ is the number $diam(A)$ = $sup${$d(a, b) : a, b ∈ A$}.
a) Suppose that $X$ is a complete and $A1 ⊃ A2 ⊃ A3 ⊃ · · · An ⊃ · · ·$ is a nested sequence of closed subsets of $X$. Prove, that if $diam (An )$ → $0$ then there is a unique point $a$ such that $a ∈ An$ for every $n$.
b) Give an example of a sequence $A1 ⊃ A2 ⊃ A3 ⊃ · · · An ⊃ · · ·$ of closed subsets of $\mathbb{R}$ such that $diam (An ) ⇸ 0$ and $\bigcap{_n}$ $An = ∅$
c) b) Give an example of a sequence $A1 ⊃ A2 ⊃ A3 ⊃ · · · An ⊃ · · ·$ of open subsets of $\mathbb{R}$ such that $diam (An ) → 0$ but $\bigcap{_n}$ $An = ∅$