Intersection of a sequence of non empty nested metric subspaces

Question

Let $X$ be a metric space

Let $\{A_n\}_{n \in \mathbb{N}} \subset X$ be a sequence of non empty nested subspaces of $X$ i.e. $A_{n+1} \subseteq A_n$

Let $$ A = \bigcap_{n=1}^\infty \overline{A_n} $$ ($\overline{A_n}$ denotes the closure of $A_n$)

My question is if is it true that: $ \forall a \in A : \forall n \in \mathbb{N} \text{ } \exists \text{ } a_n \in A_n $ such that $$ \lim_{n \to \infty} a_n = a $$

thanks.

Answer 1

Yes, it is true that each $a\in A$ is the limit of a sequence $(a_n)_{n\in\mathbb N}$, with $(\forall n\in\mathbb N):a_n\in A_n$. Take $a\in A$. For each $n\in\mathbb N$, $a\in\overline{A_n}$. So, you can take a $a_n\in A_n$ such that $d(a,a_n)<\frac1n$ and this condition assures that $\lim_{n\to\infty}a_n=a$.