The limit of a sequence of $\sup X_n$ where each $X_n$ is a real subset

Question

Let $M \geq 0$ be real positive number and let $\{X_n\}_{n \in \mathbb{N}}$ be a sequence of subsets of $\mathbb{R}$ such that:

$\forall n \in \mathbb{N}:$ $$ \forall x \in X_n : 0 \leq x \leq M $$ $$ X_{n+1} \subseteq X_n $$ $$ \bigcap_{n=1}^\infty X_n = \{0\} $$ I would like to know if is it true that: $$ \lim_{n \to \infty} (\sup X_n) = 0 $$

Thanks.

Answer 1

No, this is not necessarily true. Consider for example $$ X_n = \{0 \} \cup \{1 - \frac{1}{k} \mid k \in \mathbb N \text{ has a least } n \text{ prime factors} \}. $$