Nested Interval Theorem

Nested Interval Theorem: Let $I_n=[a_n,b_n]$ be closed intervals in $\mathbb{R}$ such that $I_{n+1}\subseteq I_n \quad \text{for all } n,$ and $\lim_{n\to\infty}(b_n-a_n)=0$. Then $\bigcap_{n=1}^\infty I_n = \{x\}$ for a unique $x\in\mathbb{R}$.

This is a concrete expression of completeness and is frequently used to construct real numbers by successive approximation.

Proof sketch (optional): The sequence $(a_n)$ is increasing and bounded above by $b_1$, so it converges to $x=\sup\{a_n\}$. Similarly, $(b_n)$ decreases and has the same limit because $b_n-a_n\to 0$. Then $x\in I_n$ for all $n$, and uniqueness follows from the shrinking lengths.