Prove lim(x sub n) = 0 if and only if lim(|x sub n|)= 0.

I am not sure how to even start this or what property of limits apply here.

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- Oct 10th 2008, 02:46 PMhayter221Limit Proof
Prove lim(x sub n) = 0 if and only if lim(|x sub n|)= 0.

I am not sure how to even start this or what property of limits apply here. - Oct 10th 2008, 03:06 PMPlato
- Oct 10th 2008, 06:40 PMhayter221
Can you show me how to incorporate this into the proof? I have that for every epsilon>0 there exists a natural number K(epsilon) such that for all n>or equal to K(epsilon) the tems x sub n satisfy |x sub n -0|< epsilon. From what you said |x sub n| = |x sub n -0| but I am not sure how this leads to lim(|x sub n|) = 0 from lim(x sub n)=0 or vice versa.

- Oct 11th 2008, 05:22 AMkalagota
well, by assumption, $\displaystyle \lim x_n = 0$ iff for every $\displaystyle \varepsilon>0$ there exists a natural number $\displaystyle K$ such that for all $\displaystyle n \geq K$ we have $\displaystyle |x_n - 0| < \varpesilon$..

and from plato's post, the last in the equality is less than epsilon..

in the same manner, $\displaystyle \lim |x_n| = 0$ iff for every $\displaystyle \varepsilon>0$ there exists a natural number $\displaystyle K$ such that for all $\displaystyle n \geq K$ we have $\displaystyle ||x_n| - 0| < \varpesilon$..