Introduction
Convergence of sequences is a fundamental concept in mathematics, particularly in real analysis. While monotonic sequences (sequences that are either strictly increasing or decreasing) have a straightforward method to prove their convergence, non-monotonic sequences require a more nuanced approach. In this article, we will delve into the steps to prove the convergence of a non-monotonic sequence using the Bolzano-Weierstrass theorem. Let's explore how to apply this theorem to demonstrate convergence in a practical scenario.

Understanding Non-Monotonic Sequences

A non-monotonic sequence is a sequence that does not consistently increase or decrease. That is, the terms of the sequence switch between increasing and decreasing. Unlike monotonic sequences, non-monotonic sequences require a different strategy for proving convergence.

Using the Bolzano-Weierstrass Theorem

The Bolzano-Weierstrass theorem provides a powerful tool for proving convergence. It states that every bounded sequence in the real numbers has a convergent subsequence. This theorem is particularly useful for non-monotonic sequences as it guarantees the existence of a convergent subsequence without requiring the sequence to be monotonic.

Steps to Prove the Convergence of a Non-Monotonic Sequence

Step 1: Show Boundedness

To apply the Bolzano-Weierstrass theorem, the first step is to show that the sequence is bounded. This means that there exist real numbers (M) and (m) such that (m leq a_n leq M) for all (n).

If the sequence is not bounded, it does not converge. Therefore, establishing boundedness is a crucial initial step.

Step 2: Apply the Bolzano-Weierstrass Theorem

Since the sequence (a_n) is bounded, the Bolzano-Weierstrass theorem guarantees the existence of at least one subsequence (a_{n_k}) that converges to some limit (L).

Step 3: Evaluate the Original Sequence

To prove that the entire sequence (a_n) converges to (L), it is often necessary to show that the original sequence approaches (L) as (n) increases. This can typically be done by demonstrating that for any (varepsilon > 0), there exists an integer (N) such that for all (n geq N), (|a_n - L|

Example: Converging Non-Monotonic Sequence

Consider the non-monotonic sequence defined as (a_n frac{-1^n}{n}).

Show Boundedness

The sequence is bounded because (-1 leq frac{-1^n}{n} leq 1) for all (n geq 1).

Apply Bolzano-Weierstrass

Since the sequence is bounded, there exists at least one convergent subsequence. For instance, the subsequence for even (n) converges to (0) as (frac{1}{2n} to 0) and the subsequence for odd (n) converges to (0) as (-frac{1}{2n 1} to 0).

Evaluate the Original Sequence

From the above, we can conclude that the entire sequence converges to (0): (lim_{n to infty} a_n 0).

Conclusion

In summary, to prove the convergence of a non-monotonic sequence, you should demonstrate that the sequence is bounded and then apply the Bolzano-Weierstrass theorem to find a convergent subsequence. Finally, you need to show that the original sequence converges to the same limit as that of the subsequence. By following these steps, you can establish the convergence of a non-monotonic sequence.