Real numbers/Nested intervals/Point/Fact

Theorem about nested intervals

Suppose that ${}I_{n}$, ${}n\in \mathbb {N}$, is a sequence of nested intervals in ${}\mathbb {R}$.

Then the intersection

$\bigcap _{n\in \mathbb {N} }I_{n}$

contains exactly one point

${}x\in \mathbb {R}$.

Nested intervals determine a unique real number.

Write a proof