The nested interval theorem

This animated video demonstrates the proof of the Nested Interval Theorem using the Monotone Convergence Principle. It begins by constructing a sequence of nested closed intervals on a number line. The narration explains that the left endpoints form a monotonically increasing sequence bounded above, while the right endpoints form a monotonically decreasing sequence bounded below. According to the Monotone Convergence Principle, both sequences must converge to limits. By showing that the length of the intervals approaches zero, it is deduced that these two limits are equal, establishing the existence of a unique common point $\xi$. Finally, the uniqueness of this point is proven via contradiction: assuming another distinct point $\eta$ exists leads to a logical conflict with the shrinking nature of the intervals.

Before you watch

• Real analysis basics
• Definition of sequence limits
• Properties of inequalities