Non-Increasing Sequences and Convergence in Real Analysis

Understanding the behavior of non-increasing sequences is fundamental in real analysis. In this article, we will explore a scenario where a non-increasing sequence ({S_n}) that is bounded below is examined to determine its convergence. We will also discuss a counterexample to illustrate that the sequence ({-1^n}) is a non-increasing sequence that is bounded below and not convergent.

What is a Non-Increasing Sequence?

A non-increasing sequence, also known as a non-negative decreasing sequence or a monotonically non-increasing sequence, is a sequence in which each element is not greater than the previous one. Formally, for a sequence ({S_n}), if (S_{n 1} leq S_n) for all (n in mathbb{N}), then ({S_n}) is a non-increasing sequence.

Bounded Below Sequences

A sequence ({S_n}) is said to be bounded below if there exists a real number (m) such that (S_n geq m) for all (n in mathbb{N}). The greatest lower bound (or infimum) of the sequence, denoted by (m), is the largest number with this property. The infimum (m) is the lower bound of the sequence that is closer to the sequence than any other lower bounds.

Convergence of Non-Increasing Sequences Bounded Below

A well-known theorem in real analysis states that a non-increasing sequence that is bounded below converges to its greatest lower bound. Formally, if ({S_n}) is a non-increasing sequence and there exists (m in mathbb{R}) such that (S_n geq m) for all (n in mathbb{N}), then ({S_n}) converges to (m).

Proof of the Convergence Theorem

Let's formulate the proof step by step.

Define (m inf{S_n : n in mathbb{N}}). This (m) is the greatest lower bound of the sequence, meaning that for all (n in mathbb{N}), we have (m leq S_n).

By the definition of the infimum, for any (epsilon > 0), there exists some (k in mathbb{N}) such that (m - epsilon

Since ({S_n}) is a non-increasing sequence, for all (n geq k), we have (S_n leq S_k).

Combining the inequalities, for all (n geq k), we have (m - epsilon

This shows that for any (epsilon > 0), there exists a natural number (k) such that for all (n geq k), (|S_n - m|

Counterexample: A Non-Increasing Sequence Not Convergent

A non-increasing sequence can be bounded below and oscillatory, which means it does not converge to a single limit. A classic example is the sequence ({-1^n}).

The sequence ({-1^n}) is defined as follows:

(-1^n -1) if (n) is odd

(-1^n 1) if (n) is even

Clearly, ({-1^n}) is a non-increasing sequence since for odd (n), (-1^n

However, ({-1^n}) does not converge because it alternates between (-1) and (1). Hence, it does not satisfy the condition of convergence to a single limit value.

Conclusion

In summary, a non-increasing sequence that is bounded below is convergent. The sequence ({-1^n}) serves as a counterexample where a non-increasing sequence is bounded below but does not converge, demonstrating the importance of additional conditions in real analysis.

Further Readings and Resources

For further study, you can refer to the following resources:

Wikipedia: Sequence (Mathematics)

Math is Fun: Sequences and Series

Understanding these concepts will enhance your grasp of real analysis and sequence theory in mathematics.