Lecture 9: Limsup, Liminf, and the Bolzano-Weierstrass Theorem

TL;DR

Every bounded sequence has a convergent subsequence if and only if the limsup equals the liminf.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: All right, so we proved these two theorems last time, and we used them for-- and we had a couple of applications of them. So the first theorem, simple theorem, was that a sequence converges to x if and only if the limit as n goes to infinity of the absolute value of xn minus x goes to 0. And then w... Read More

Key Insights

• ❓ Limsup and liminf are well-defined and exist for bounded sequences.
• ↗️ The limsup of a sequence gives an upper bound on the elements of the sequence.
• ❓ Limsup and liminf can be used to determine if a sequence converges.

Questions & Answers

Q: What are the limsup and liminf of a sequence?

The limsup is the limit as n goes to infinity of the supremum of the set of elements of the sequence from n onwards. The liminf is the limit as n goes to infinity of the infimum of the set of elements of the sequence from n onwards.

Q: How do you prove that limsup and liminf exist for bounded sequences?

By showing that the sequences of suprema and infima are monotone and bounded, we prove that the limsup and liminf exist for bounded sequences.

Q: What is the Bolzano-Weierstrass theorem?

The Bolzano-Weierstrass theorem states that every bounded sequence has a convergent subsequence.

Q: How can we use limsup and liminf to determine if a sequence converges?

If the limsup equals the liminf, then the sequence converges. If the sequence converges, the limsup and liminf will also equal the limit of the sequence.

Summary & Key Takeaways

• The limsup and liminf are well-defined for bounded sequences.

• Limsup and liminf sequences are monotone and bounded.

• The limsup and liminf exist for every bounded sequence, and there exist subsequences that converge to them.

• The limsup equals the liminf if and only if the original sequence converges.