Monotone Sequences: Proof & Strictly Increasing?

Thread starter mathanon
Start date Sep 27, 2013
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Sequences

In summary, a monotone sequence is a sequence of numbers that either constantly increases or decreases, with each term being greater than or equal to the previous term (for non-decreasing sequences) or less than or equal to the previous term (for non-increasing sequences). A strictly increasing sequence always moves upwards, while a non-decreasing sequence can sometimes stay at the same level. To prove that a sequence is strictly increasing, one must show that each term is greater than the previous term. Monotone sequences cannot be both strictly increasing and strictly decreasing. The concept of monotone sequences is important in mathematics because they help us understand the behavior of a sequence, have applications in various fields of math, and can simplify complex problems in proofs.

Sep 27, 2013

mathanon

Let A be a nonempty subset of R that is bounded above and let α=supA. Show that there exists a monotone increasing sequence {an} in A such that α=lim an. Can the sequence {an} be chosen to be strictly increasing?

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Sep 27, 2013

HallsofIvy

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Homework Helper

For every positive n, look at the set [itex]\{ a_i| |a_i- a|<1/n\}[/itex]. Can you see that this set is non-empty for all n? Choose a member of this set to be in the subsequence.

FAQ: Monotone Sequences: Proof & Strictly Increasing?

1. What is a monotone sequence?

A monotone sequence is a sequence of numbers that either constantly increases or decreases. In other words, each term in the sequence is either greater than or equal to the previous term (in the case of a non-decreasing sequence) or less than or equal to the previous term (in the case of a non-increasing sequence).

2. What is the difference between a strictly increasing and a non-decreasing sequence?

A strictly increasing sequence is one where each term is greater than the previous term, while a non-decreasing sequence allows for terms to be equal to the previous term. In other words, a strictly increasing sequence is always moving upwards, while a non-decreasing sequence can sometimes stay at the same level.

3. How do you prove that a sequence is strictly increasing?

To prove that a sequence is strictly increasing, you must show that each term is greater than the previous term. This can be done through various methods, such as induction or using the definition of a monotone sequence. You can also prove that a sequence is strictly increasing by showing that the difference between consecutive terms is always positive.

4. Can a monotone sequence be both strictly increasing and strictly decreasing?

No, a monotone sequence can only be strictly increasing or strictly decreasing. This is because a strictly increasing sequence cannot have terms that are less than the previous term, and a strictly decreasing sequence cannot have terms that are greater than the previous term. Therefore, a sequence cannot satisfy both conditions simultaneously.

5. Why is the concept of monotone sequences important in mathematics?

Monotone sequences are important in mathematics because they help us understand the behavior of a sequence and the values it can take on. They also have many applications in different areas of mathematics, such as in the study of limits, derivatives, and integrals. Additionally, monotone sequences are often used in proofs and can help simplify complex problems by reducing them to a simpler case.