Nested interval theorem

For $a_n\le b_n$ define a sequence of intervals by $I_n=[a_n,b_n].$ If $I_{n+1}\subseteq I_n$ and ${\lim }_{n\to \infty }{b}_n-a_n=0$, then there is exactly one point $x\in \mathbb{R}$ that is in each interval. Written prettier:

${\bigcap }_{n=0}^{\infty }{I}_n=\{x\}$

And $\{a_n\}\to x$, $\{b_n\}\to x$.

Proof