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Chapter 3: Problem 11

Suppose that \(x_{n} \geq 0\) for all \(n \in \mathbb{N}\) and that \(\lim \left((-1)^{n} x_{n}\right)\) exists. Show that \(\left(x_{n}\right)\) converges.

Short Answer

Expert verified
The provided sequence \(x_{n}\) must converge because it is a combination of two subsequences that are both bounded and all its terms are non-negative.

Step by step solution

01

Understanding the Problem

According to the data, \(\lim \left((-1)^{n} x_{n}\right)\) exists. This implies that the sequence \(-1^n x_n\) is bounded. The sequence \(x_n\) is sequence of non-negative terms and hence its limit will exist only if it is bounded.
02

Constructing Two Sub-Sequences

To prove the convergence of \(x_{n}\), we consider two subsequences. The first subsequence \(x_{2n}\) consists of even numbered terms and the second subsequence \(x_{2n+1}\) consists of odd numbered terms. Since the limit of \((-1)^{n} x_{n}\) exists, both subsequences are bounded.
03

The Sequence is Bounded

From the previous step, we have shown that both \(x_{2n}\) and \(x_{2n+1}\) are bounded. The original sequence \(x_n\), which is a combination of these two subsequences, must also be bounded since both subsequences are bounded.
04

Connecting the Limits of the Subsequences

By the theorem that states a sequence is convergent if and only if all its subsequences converge to the same limit, since all the terms \(x_n\) are non-negative and its subsequences are bounded, the sequence \(x_n\) itself must converge.

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