Lecture 7: Convergent Sequences of Real Numbers

TL;DR

Convergent sequences have limits, and subsequences of a convergent sequence also converge to the same limit.

Transcript

[SQUEAKING] [RUSTLING] [CLICKING] CASEY RODRIGUEZ: OK, so last time, we were talking about sequences, and I introduced the notion of a limit of a sequence. So we say that x n converges to x if for all epsilon positive there exists an M, a natural number, such that for all little n bigger than or equal to capital M we have x sub n minus x is less th... Read More

Key Insights

• ⛔ Convergent sequences have limits, while non-convergent sequences diverge or oscillate.
• 👍 The negation of convergence can be used to prove that a sequence does not converge to a specific limit.
• ❓ Monotone sequences satisfy the converse of the theorem that every convergent sequence is bounded.
• 🪈 Subsequences are obtained by selecting entries from an original sequence in an increasing order.

Questions & Answers

Q: What is the definition of a convergent sequence?

A convergent sequence is a sequence that has a limit, meaning the entries of the sequence get arbitrarily close to a single value as the sequence progresses.

Q: What is the negation of convergence?

The negation of convergence is when a sequence does not converge to the specified limit. This occurs when there exists a "bad epsilon" value such that the entries of the sequence do not get within the tolerance defined by epsilon.

Q: What is a bounded sequence?

A bounded sequence is a sequence that does not increase indefinitely. It stays within a certain range, which means there is a maximum and minimum value it can reach.

Q: When does a sequence not converge to a specific limit?

A sequence does not converge to a specific limit if, for any given "bad epsilon" value, there exists an entry in the sequence that is far away from the limit. This violates the definition of convergence.

Q: What are monotone sequences?

Monotone sequences are either increasing or decreasing sequences. Increasing sequences have entries that get larger as the sequence progresses, while decreasing sequences have entries that get smaller. Both types of sequences satisfy the converse of the theorem that states every convergent sequence is bounded.

Q: What is a subsequence?

A subsequence is obtained by selecting specific entries from an original sequence according to an increasing sequence of natural numbers. Each chosen entry represents the subsequence, and the order of the selected entries follows the increasing sequence.

Q: Do subsequences also converge to the same limit?

Yes, subsequences of a convergent sequence also converge to the same limit as the original sequence. If the original sequence converges to x, then all subsequences will also converge to x.

Q: What happens if a sequence is not convergent?

If a sequence is not convergent, it either diverges or oscillates between one or more values. This means the entries of the sequence do not get closer and closer to a single value.

Summary & Key Takeaways

• A convergent sequence is a sequence that has a limit, indicated as x sub n arrow x.

• The negation of convergence is when a sequence does not converge to a given x. In this case, there exists a bad epsilon that violates the definition of convergence.

• A bounded sequence is a sequence that does not increase indefinitely, and a convergent sequence is always bounded.

• Monotone sequences, either increasing or decreasing, are sequences that satisfy the converse of the theorem that states every convergent sequence is bounded.

• Subsequences are obtained by selecting specific entries from an original sequence, as long as the chosen entries are in increasing order.