Question #93339
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.
1
Expert's answer
2019-08-26T10:35:16-0400

limnXn=0ε>0 N n>N Xn0<ε.\lim\limits_{n\to\infty} X_n=0 \leftrightarrow \forall \varepsilon>0 \ \exists N \ \forall n>N \ |X_n-0|<\varepsilon.

limnXn=0ε>0 N n>N Xn0<ε.\lim\limits_{n\to\infty} |X_n|=0 \leftrightarrow \forall \varepsilon>0 \ \exists N \ \forall n>N \ \bigl||X_n|-0\bigr|<\varepsilon.

Since Xn0=Xn0|X_n-0|=\bigl||X_n|-0\bigr| , we have limnXn=0limnXn=0\lim\limits_{n\to\infty} X_n=0 \Leftrightarrow \lim\limits_{n\to\infty} |X_n|=0 .

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


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS