Nested intervals/Sequence/Convergence/Exercise

Let ${}I_{n}$, ${}n\in \mathbb {N}$, be a sequence of nested intervals in ${}\mathbb {R}$ and let ${}{\left(x_{n}\right)}_{n\in \mathbb {N} }$ be a real sequence with ${}x_{n}\in I_{n}$ for all ${}n\in \mathbb {N}$. Prove that this sequence converges to the unique number belonging to the intersection of the family of nested intervals.

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