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May 14th, 2017, 11:51 PM   #1
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Convergence of a sub sequence

Hello
Any one help me with this!!!

If ${a_n }$ is a sequence converging to $l$. Let $b_n$ = $a_2n$, if n is odd,
$a_3n$, if n is even.
Then the sequence ${b_n}$

A. need not converge
B. should converge to $0$.
C. should converge to $2l$ or to $3l$.
D. should converge to $l$.

I know that if a sequence is convergent then its subsequents also converge.
But I dont know whether they converge to the same limit as that of the original sequence or not ?
Please help!!!
thanks in advance
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May 15th, 2017, 03:24 AM   #2
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Yes, of course they have to converge to the same limit! If the sequence, , converges to x, that means, for any and sufficiently large n, . Now, suppose some subsequence, converged to some other number, y. That would mean that, for some and sufficiently large n, . Take and realize that those cannot both be true!
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May 15th, 2017, 03:39 AM   #3
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Quote:
Originally Posted by Country Boy View Post
Yes, of course they have to converge to the same limit! If the sequence, , converges to x, that means, for any and sufficiently large n, . Now, suppose some subsequence, converged to some other number, y. That would mean that, for some and sufficiently large n, . Take and realize that those cannot both be true!
Thank you so much for the clarification
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