Decreasing Sequence

Definition

A decreasing sequence is a sequence of numbers where each term is less than or equal to the preceding term. This means that for a sequence \( (a_n) \), it holds that \( a_n \geq a_{n+1} \) for all \( n \). Understanding decreasing sequences is important because they relate closely to concepts like convergence, boundedness, and the relationship between monotone sequences and Cauchy sequences.

5 Must Know Facts For Your Next Test

1. A decreasing sequence can converge to a limit, meaning that as the terms progress, they approach a particular value.
2. If a decreasing sequence is bounded below, it must converge to its greatest lower bound (infimum).
3. Decreasing sequences are helpful in proving properties related to Cauchy sequences since Cauchy sequences are always bounded.
4. In a decreasing sequence, if any two terms are compared, the earlier term will always be greater than or equal to the later term.
5. The property of being decreasing can help establish whether certain limits exist and facilitate understanding convergence in sequences.

Review Questions

• How does a decreasing sequence relate to the concepts of boundedness and convergence?
  • A decreasing sequence is connected to boundedness because if it is bounded below, it must converge. The property of being bounded below ensures that there is a greatest lower bound (infimum) that the terms of the sequence will approach as they decrease. This convergence is crucial for understanding how sequences behave over time and helps in establishing their limits.
• In what way do decreasing sequences serve as examples when discussing Cauchy sequences?
  • Decreasing sequences serve as clear examples in the context of Cauchy sequences because they illustrate how sequences can be bounded and how this leads to convergence. Since Cauchy sequences must have their terms becoming arbitrarily close to each other as the sequence progresses, a decreasing sequence that is also Cauchy demonstrates these properties explicitly. Such examples make it easier to visualize the concepts of convergence and boundedness together.
• Evaluate how understanding decreasing sequences contributes to analyzing the properties of monotonic and Cauchy sequences.
  • Understanding decreasing sequences enhances the analysis of monotonic and Cauchy sequences by providing concrete examples that illustrate key principles. For instance, recognizing that all decreasing sequences are monotonic helps clarify their behavior. Furthermore, since every Cauchy sequence is bounded and has the potential to converge, seeing how decreasing sequences fulfill these criteria enriches our comprehension of why certain limits exist. This evaluation shows the interconnectedness of these concepts in mathematical analysis.

Related terms

Monotonic Sequence: A sequence that is either entirely non-increasing or non-decreasing. A decreasing sequence is a specific type of monotonic sequence.

Cauchy Sequence: A sequence where for every positive number \( \, \epsilon \), there exists an integer \( N \) such that for all integers \( m, n > N \), the terms of the sequence are close to each other, regardless of their position in the sequence.

Bounded Sequence: A sequence that has both an upper and lower bound, meaning all its terms fall within a specific range. A decreasing sequence can be bounded above.