# Question About Nested Interval Property

This is a theorem from Abott's Real Analysis textbook.

Theorem 1.4.1 (Nested Interval Property)

For each $$n\in N,$$ assume we are given a closed interval $$I_n = [a_n,b_n] = \{x\in R : a_n\leq x\leq b_n \}$$. Assume also that each $$I_n$$ contains $$I_n{_+}{_1}$$. Then the resulting sequence of closed intervals

$$I_1\subseteq I_2\subseteq I_3\subseteq I_4\subseteq \cdots$$

has a nonempty intersection; that is, $$\bigcap\limits_{n=1}^{\infty} I_{n} \neq \emptyset$$

QUESTION HERE:

Wikipedia says that an given an interval $$I_n$$ and $$I_n{_+}{_1}$$, $$I_n{_+}{_1}$$ is a subset of $$I_n$$ which makes total sense. But in the theorem it reads that the "resulting sequence of closed intervals" and below it says $$I_1$$ is a subset of $$I_2$$, so on and so forth, which is not true. Am I misreading this?

• If you have quoted the theorem exactly, you are right and Abbot's book has a typo in that all the containment symbols should be reversed to agree with what is said verbally (and with other statements I've seen of nested interval theorem). May 18, 2021 at 23:32
• @coffeemath The OP's quote was actually not exact: see screenshot below. Feb 12 at 14:29

The book says $$I_1 ⊇ I_2 ⊇ I_3 ⊇ I_4 ⊇···$$ (you have it flipped)