Lebl-Basic-Analysis-I · Problem 2.1.7

Sequences and Series

Mathematics · sequences-and-limits

Problem

Let {xn}n=1∞\{x_n\}_{n=1}^\infty be a sequence. a) Show that lim⁡n→∞xn=0\lim_{n\to\infty} x_n = 0 (that is, the limit exists and is zero) if and only if lim⁡n→∞∣xn∣=0\lim_{n\to\infty} |x_n| = 0. b) Find an example such that {∣xn∣}n=1∞\{|x_n|\}_{n=1}^\infty converges and {xn}n=1∞\{x_n\}_{n=1}^\infty diverges.
Intuitive explanation
PART (A)
This is about understanding what "convergence to zero" really means—it's symmetric around zero, so the sign shouldn't matter. The key insight is that the distance from xnx_n to zero is exactly ∣xn∣|x_n|, so requiring xnx_n to get arbitrarily close to zero is the same as requiring ∣xn∣|x_n| to get arbitrarily close to zero. You'll use the ϵ\epsilon-NN definition in both directions.
PART (B)
You need the absolute values to settle down to a limit while the original sequence bounces around and never converges. Think about what happens if you take a convergent sequence of positive numbers and randomly flip the signs—the absolute values still converge, but the sequence itself might oscillate forever.
Intuitive explanation
PART (A)
This is about understanding what "convergence to zero" really means—it's symmetric around zero, so the sign shouldn't matter. The key insight is that the distance from xnx_n to zero is exactly ∣xn∣|x_n|, so requiring xnx_n to get arbitrarily close to zero is the same as requiring ∣xn∣|x_n| to get arbitrarily close to zero. You'll use the ϵ\epsilon-NN definition in both directions.
PART (B)
You need the absolute values to settle down to a limit while the original sequence bounces around and never converges. Think about what happens if you take a convergent sequence of positive numbers and randomly flip the signs—the absolute values still converge, but the sequence itself might oscillate forever.

Related theory

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