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May 15th, 2017, 12:51 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2  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 
May 15th, 2017, 04:24 AM  #2 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,922 Thanks: 785 
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!

May 15th, 2017, 04:39 AM  #3  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 211 Thanks: 2  Quote:
 

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