nested interval theorem

Picture an endless sequence of closed intervals, each one sitting inside the previous, like nested boxes closing in on a target. The nested interval theorem promises that there is always at least one point caught inside every box at once, so the boxes never close on empty space.

Precisely: if I_1 contains I_2 contains I_3 and so on, where each I_n is a closed bounded interval, then the intersection of all the I_n is nonempty. If in addition the lengths of the intervals shrink to zero, the intersection is a single point, and that point is the common limit of both endpoint sequences.

This theorem is one of the equivalent faces of completeness; over the reals it follows from the least upper bound property, taking the common point to be the supremum of the left endpoints. It is the engine behind many existence proofs, such as the bisection proof of the intermediate value theorem and one proof of the Bolzano–Weierstrass theorem.

Two hypotheses are essential and cannot be dropped. The intervals must be closed: the open intervals from 0 to 1 over n nest and shrink, yet their intersection is empty because the would-be common point 0 is excluded from each. And the intervals must be bounded: the rays from n to infinity nest but share no point.