Andromeda
Note

Monotonic Sequence Theorem

Definition

The Monotonic Sequence Theorem states that every bounded, monotonic sequence is convergent. That is, if a sequence is either purely increasing or purely decreasing, and it cannot exceed a certain value (the bound), it must approach a finite limit.

Why It Matters

This theorem provides a guarantee of convergence without needing to know the limit’s value beforehand. In numerical analysis and computer science, this certainty prevents infinite loops and ensures that iterative algorithms will eventually settle on a stable result.

Core Concepts

  • Monotonicity: The sequence must always move in one direction (either an≤an+1a_n \leq a_{n+1} or an≥an+1a_n \geq a_{n+1}).

    • How to read: “The nth term of the sequence is less than or equal to the n plus first term; or the nth term is greater than or equal to the n plus first term.”
    • Meaning: Increasing: each term is at least as large as the previous; decreasing: each term is at least as small—no oscillation in either case.

  • Boundedness:

    • If increasing, it must be bounded above (an≤Ma_n \leq M).

      • How to read: “The nth term of the sequence is less than or equal to the upper bound M.”
      • Meaning: An upper ceiling MM prevents the sequence from growing without bound.

    • If decreasing, it must be bounded below (an≥ma_n \geq m).

      • How to read: “The nth term of the sequence is greater than or equal to the lower bound m.”
      • Meaning: A lower floor mm prevents the sequence from decreasing without bound.

  • Convergence: The theorem guarantees that the limit exists, though it does not provide a formula to calculate it. The limit is the least upper bound (supremum) for increasing sequences or the greatest lower bound (infimum) for decreasing sequences.

Connected Concepts

Connected notes