Covering-interval argument for the existence of cycles

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Problem X.1.29 from Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. "Избранные задачи по вещественному анализу" [МГЛ+04]More problems in Section X.1 "Topological Dynamics"

Let ff be a continuous map of a segment Δ\Delta into itself, and let I0,I1,…I_{0}, I_{1}, \ldots be a finite or infinite sequence of segments, with I0⊂Δ,In+1⊂f(In)I_{0} \subset \Delta , I_{n+1} \subset f\left(I_{n}\right) for every n⩾0n \geqslant 0. Prove that

?
(a)

there exists a sequence of segments K0⊃K1⊃K2⊃⊃K3⊃…,K0⊂I0K_{0} \supset K_{1} \supset K_{2} \supset \supset K_{3} \supset \ldots , K_{0} \subset I_{0}, such that fn(Kn)=In(n⩾0)f^{n}\left(K_{n}\right)=I_{n}(n \geqslant 0);

(b)

there exists a point x∈Δx \in \Delta for which xn=fn(x)∈Inx_{n}=f^{n}(x) \in I_{n} for every nn.

References

[МГЛ+04] Макаров Б.М., Голузина М.Г., Лодкин А.А., Подкорытов А.Н. Избранные задачи по вещественному анализу. 2-е изд., Невский Диалект; БХВ-Петербург, 2004.