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- March 27th 2013, 08:30 PMbastianteReal Analysis, can someone help me?
- March 28th 2013, 12:03 AMhollywoodRe: Real Analysis, can someone help me?
If converges, it is obvious that converges, right?

Suppose that does not converge to x. What does that mean in terms of N and ? Does that give you an idea for the sequence ?

- Hollywood - March 28th 2013, 03:11 AMMINOANMANRe: Real Analysis, can someone help me?
Bastiante study this theorem

Theorem

If a sequence ( Xn ) of real numbers converges to a real number x then any subsequence of X also converges to x.

Proof:

Let ε >0 be given and let K(ε) be such that if n>= K(ε) , then | xn - x|<ε . If r1<r2<r3…<rn <… is an increasing sequence of natural numbers, it can easily be proven by induction that rn>=n . Hence , if n >=K(ε) we also have rn>=n>=K(ε) so that |xrn – x| <ε . This proves that the subsequence ( Xrn ) also converges to x , and the theorem is proved.

MINOAS