Answer to Question #93339 in Real Analysis for Andy Reddy

Question

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.

Answer 1

$\lim\limits_{n\to\infty} X_n=0 \leftrightarrow \forall \varepsilon>0 \ \exists N \ \forall n>N \ |X_n-0|<\varepsilon.$

$\lim\limits_{n\to\infty} |X_n|=0 \leftrightarrow \forall \varepsilon>0 \ \exists N \ \forall n>N \ \bigl||X_n|-0\bigr|<\varepsilon.$

Since $|X_n-0|=\bigl||X_n|-0\bigr|$ , we have $\lim\limits_{n\to\infty} X_n=0 \Leftrightarrow \lim\limits_{n\to\infty} |X_n|=0$ .

Consider sequence $X_n=(-1)^n.$ It is divergent one, but sequence $|X_n|=1$ is convergent.