Strictly increasing

Definition

A sequence is considered strictly increasing if each term is greater than the preceding term. This means that for a sequence {a_n}, it holds true that if n < m, then a_n < a_m. This characteristic indicates that as you move through the sequence, the values are continuously rising without any repeats or plateaus, which has important implications for convergence and limits.

5 Must Know Facts For Your Next Test

1. In a strictly increasing sequence, every term is unique, meaning no two terms can be equal.
2. Strictly increasing sequences are useful in proofs and analyses involving limits, as they help establish certain bounds.
3. If a strictly increasing sequence is bounded above, it must converge to its least upper bound (supremum).
4. The concept of strictly increasing sequences is often used in real analysis to demonstrate properties of functions and continuity.
5. Every strictly increasing sequence is also monotonic, but not every monotonic sequence is strictly increasing.

Review Questions

• What properties distinguish a strictly increasing sequence from other types of sequences?
  • A strictly increasing sequence has the distinct property that each term is greater than the one before it. Unlike non-decreasing sequences where terms can be equal, strictly increasing sequences ensure that all values are unique. This uniqueness plays a crucial role in analyzing limits and convergence behaviors of sequences.
• How does being strictly increasing affect the convergence of a sequence?
  • If a strictly increasing sequence is bounded above, it must converge to its least upper bound or supremum. The strict nature of the increase ensures that the terms approach this supremum without ever actually reaching it. This property is vital in real analysis, as it helps establish how limits operate within sequences and the behavior of functions.
• Evaluate the significance of strictly increasing sequences in understanding the behavior of real-valued functions.
  • Strictly increasing sequences are significant because they provide insight into how real-valued functions behave. When a function's values form a strictly increasing sequence, it indicates that the function is moving upward without any plateaus or decreases over its domain. This property assists in determining the function's monotonicity and aids in establishing whether limits exist or how they can be evaluated, which are essential concepts in mathematical analysis.

Related terms

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

Convergence: The property of a sequence approaching a specific value as the index goes to infinity. Strictly increasing sequences can converge under certain conditions.

Bounded Sequence: A sequence that has both upper and lower bounds. A strictly increasing sequence can be bounded, affecting its convergence behavior.