Help with nested interval proof

The proof I'm working on states:

Let $\displaystyle [a_{n},b_{n}]$ be nested intervals such that $\displaystyle b_{n}-a_{n}$ is null. Let $\displaystyle x_{0}$ be the real number satisfying $\displaystyle \bigcap_{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}$. Let $\displaystyle \delta >0$. Prove there is an $\displaystyle N$, such that $\displaystyle [a_{N},b_{N}]\subseteq (x_{0}-\delta ,x_{0}+\delta )$.

Here is the direction that I think I need to go but I'm not sure...

Since $\displaystyle \bigcap_{n=1}^\infty [a_{n},b_{n}]=\{x_{0}\}$, and $\displaystyle x_{0}-\delta <x_{0}<x_{0}+\delta$ then there must be an interval $\displaystyle [a_{N},b_{N}]$ that is also contained in $\displaystyle (x_{0}-\delta ,x_{0}+\delta )$. But all that might be completely wrong.