Question

Let \(\left\\{a_{n}\right\\}\) be a monotonic sequence such that \(a_{n} \leq 1\). Discuss the convergence of \(\left\\{a_{n}\right\\} .\) If \(\left\\{a_{n}\right\\}\) converges, what can you conclude about its limit?

Short answer

The given monotonic sequence \(\{a_{n}\}\) is bounded and therefore it is convergent. The limit of the sequence \(\{a_{n}\}\) is less than or equal to 1.

Step-by-step solution

Understanding the Problem

Given a monotonic sequence \(\{a_{n}\}\) such that \(a_{n} \leq 1\). The task is to discuss the convergence of the sequence. If the sequence converges, determine the limit.

Applying Monotone Convergence Theorem

According to the Monotone Convergence Theorem, every bounded and monotonic sequence is convergent. Since \(\{a_{n}\}\) is monotonic and bounded by 1, it must be convergent.

Discussing the Convergence

Since \(\{a_{n}\}\) is a monotonic sequence and it is bounded, it is convergent. That implies the limit of the sequence exists.

Conclusion about the limit

Given that \(a_{n} \leq 1\) for all \(n\), it follows that the limit of the sequence \(\{a_{n}\}\) will not be more than 1. This is because every term in the sequence is less than or equal to 1, so the limit of the sequence will also be less than or equal to 1.