Bounded Sequences, Completeness Axiom, and the Monotonic Sequence Theorem

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2025-04-11 17:11

In this video I first go over the definition of bounded sequences, then discuss the completeness axiom in number theory and how it is used to proof the monotonic sequence theorem. A sequence is bounded above if there is a number greater than every term in the sequence. A sequence is bounded below if there is a number smaller than every term. The completeness axiom simply states that for a set of real numbers with an upper bound, then there exists a number that is the least or smallest of upper bounds. Since an infinite number of upper bounds can exists, the least upper bound is simply the smallest one. This axiom also illustrates how there are no gaps or holes in real numbers, unlike that for the sets of only irrational or only rational numbers (the combination of which simply yield the set of real numbers).

The monotonic sequence theorem states that every bounded and monotonic sequence (increasing or decreasing) are convergent. By the completeness axiom for real numbers, I rearrange the least upper bound of the sequence to obtain the definition of the limit of a sequence, thus proving its convergence.

#math #sequences #calculus #completeness #logic

12 Bounded Monotonic Sequence Theorem.jpeg

Timestamps

Definition 5: Sequences bounded above and bounded below: 0:00
Not every bounded sequence is convergent: 2:16
Not every monotonic sequence is convergent: 3:05
A bounded and monotonic sequence must be convergent: 3:25
Completeness Axiom: no hole or gap in the real number line: https://x.com/i/grok/share/X04g5xItDl2FVk8iZxLgQjfvc 5:16
Completeness of the real number Wikipedia: https://en.wikipedia.org/wiki/Completeness_of_the_real_numbers 8:13
Monotonic Sequence Theorem and proof 9:30
Used completeness axiom to obtain the definition of the limit: 14:16
Similar proof for greatest lower bound for a decreasing and bounded sequence: 14:46