Prove that lim (Xn)=0 if and only if lim(|Xn|)= 0. Give an example to show that the convergence of (|Xn|)= 0 need not imply the convergence of (Xn)= 0.
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Expert's answer
2019-08-26T10:35:16-0400
n→∞limXn=0↔∀ε>0∃N∀n>N∣Xn−0∣<ε.
n→∞lim∣Xn∣=0↔∀ε>0∃N∀n>N∣∣∣Xn∣−0∣∣<ε.
Since ∣Xn−0∣=∣∣∣Xn∣−0∣∣ , we have n→∞limXn=0⇔n→∞lim∣Xn∣=0 .
Consider sequence Xn=(−1)n. It is divergent one, but sequence ∣Xn∣=1 is convergent.
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