Nested Intervals Theorem

Last updated Nov 1, 2022

Let $[a_{n}, b_{n}]$ be a Sequence of Closed Intervals in $\mathbb{R}$ s.t. $b_{1} \geq b_{2} \geq …$ and $a_{1} \leq a_{2} \leq \cdots$ and $b_{n} - a_{n} \to 0$ as $n \to \infty$. Then $$\bigcap\limits_{n \in \mathbb{N}} [a_{n}, b_{n}] = {c}$$ for some $c \in \mathbb{R}$.