91Ó°ÊÓ

In mathematics, an **inequality** expresses the relative size or order of two values. In the context of this problem, we deal with the inequality \(|a_{n+1} - a_n| < k|a_n - a_{n-1}|\) for a sequence \((a_n)\) where \(0 < k < 1\). This inequality indicates that the differences between consecutive terms in the sequence are getting progressively smaller with each step.
This concept is vital because when a sequence has such diminishing differences, it moves towards meeting the Cauchy sequence condition. Essentially, the application of this inequality is a stepping stone to prove that the sequence will eventually balance its progression, such that it converges as it gets further along its terms.

• Thanks to this inequality, the factor \(k\) effectively contracts the difference between every consecutive pair of terms.
• As \(k\) is less than 1, this ensures that the contraction is sufficient for convergence support.
• Over multiple iterations, the effects compound, making the difference between terms increasingly small.

This inequality sets the stage for understanding how sequences like these behave and the underlying pattern that directs them towards convergence.

A **convergent sequence** is one whereby the terms approach a specific value as they progress to infinity. More formally, a sequence \((a_n)\) converges to a limit \(L\) if, for every \(\varepsilon > 0\), there exists an integer \(N\) such that for all \(n \geq N\), \(|a_n - L| < \varepsilon\).
In this exercise's realm, we're verifying the sequence to be a **Cauchy sequence**, which is a necessary condition for convergence in the real numbers.

• A key insight is that any Cauchy sequence in real numbers is also convergent.
• The shrinking differences imposed by the inequality \(|a_{n+1} - a_n| < k|a_n - a_{n-1}|\) push the sequence seamlessly towards converging.
• As shown in the solution steps, we prove \(|a_m - a_n|\) can be made arbitrarily small by appropriately picking \(m\) and \(n\), reinforcing the sequence's convergence criterion.

Thus, while \((a_n)\) under these conditions is not only defined as Cauchy, it pushes forward into **convergence**, reinforcing Cauchy’s utility as a precursor to guaranteed convergence in analytic contexts.

A **geometric series** is a series of terms where each term after the first is found by multiplying the previous one by a constant, called the common ratio. The structure of a geometric series is: \( S = a + ar + ar^2 + ar^3 + \ldots \)
In our scenario, the sequence inequality introduces us to a geometric progression through the diminishing differences. The recursive application \(|a_{n+1} - a_n| < k|a_n - a_{n-1}|\) feels more pronounced when we break it down:

• Every term generates from the prior term scaled by the factor \(k\).
• This forms a chain \( b_{n}, b_{n+1} < kb_n, b_{n+2} < k^2b_n, \) and so forth, reflecting the form of a geometric series.
• Summing this up generates \(k + k^2 + \ldots + k^{m-n}\), which converges provided \(0 < k < 1\).

The convergence of this series aids in stabilizing the sequence itself since the sum \( \frac{k}{1-k} \) grants a bound for the sequence difference \(|a_m - a_n|\). This delineation shows how diminishing geometric escalation can indeed consolidate sequence convergence, supplementing the sufficient conditions for calling a sequence **Cauchy**. By leveraging the nature of geometric series, the solution elucidates how terms can reduce adequately to prove convergence criteria.