Nested intervals/Convergence of lengths/Non empty intersection/Exercise

Give an example of a sequence of closed intervals (${}n\in \mathbb {N} _{+}$)

${}I_{n}=[a_{n},b_{n}]\subseteq \mathbb {R} \,$

such that ${}b_{n}-a_{n}$ is a null sequence, such that the intersection ${}\bigcap _{n\in \mathbb {N} _{+}}I_{n}$ consists in exactly one point, but such that ${}I_{n}$ is not a family of

nested intervals.

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