Let $(X,d)$ a metric space. Prove that the statements are equivalent :
$\textbf{1.}$ For all sequence of closed ball $\{B_n\}$ such that: $B_{n+1} \subseteq B_{n}, \forall n\in \mathbb{N}$ and their radius tend to zero we have to :
$$ \bigcap_{n \in \mathbb{N}} B_n \neq \emptyset $$
$\textbf{2.}$ For all sequence of closed sets $\{C_n\}$ such that: $C_{n+1} \subseteq C_{n}, \forall n\in \mathbb{N}$ and $diam(C_n) \rightarrow 0$, we have to :
$$ \bigcap_{n \in \mathbb{N}} C_n \neq \emptyset $$
$2) \implies 1)$ is obviously but the other implication I can´t prove. All help is well received, thanks.
1) implies 2): let $r_n$ be the diameter of $C_n$. First note that if the intersection of some subsequence of $C_n$'s is non-empty then the intersection of all the $C_n$'s is nonempty. Hence there is no loss of generality in assuming that $\sum_n r_n <\infty$. Now let $s_n=2(r_n+r_{n+1}+...)$. Let $x_n \in C_n$ for all $n$. A simple argument using triangle inequality shows that the closed balls of radii $s_n$ around $x_n$ form a decreasing sequence of closed balls whose radii tend to $0$. Let $x$ be a point in the intersection of these balls. Since $x_n \to x$ it is clear that $x$ belongs to each $C_n$.
Will be happy to add more details if needed.