May 24th, 2017, 06:52 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 232 Thanks: 2  Rational terms sequence
Let $(a_n)$ be a sequence where all rational numbers are terms (and all terms are rational). Then A) no sub sequence of $(a_n)$ converges. B) there are uncountably many convergent sub sequences of $(a_n)$. C) Every limit point of $(a_n)$ is a rational number. D) no limit point of $(a_n)$ is a rational number. This question can have multiple answers. My answers are Option B&D. Please check and let me know. Thank you 
May 24th, 2017, 07:16 PM  #2  
Senior Member Joined: Sep 2015 From: USA Posts: 2,427 Thanks: 1314  Quote:
B) is clearly true, consider $m + \dfrac 1 n,~\forall m \in \mathbb{Z}$ I believe (C) is false. The rationals are dense in the reals so for any real number you can construct a sequence of rationals that will converge to it. D) is clearly false, $\lim \limits_{n \to \infty} \dfrac 1 n = 0 \in \mathbb{Q}$  
May 24th, 2017, 07:54 PM  #3  
Senior Member Joined: Aug 2012 Posts: 2,305 Thanks: 705  If I'm understanding the question, the sequence includes all the rationals. That is, it's an enumeration of the rationals. And those are very wild, aren't they? I am not saying this isn't false ... but I don't see that it's clearly false. I'm not sure I see the proof. Maybe I'm missing something obvious but I'm wondering why some enumeration couldn't jump around so much that no subsequence converges. The Bolzanoâ€“Weierstrass theorem says that every bounded sequence has a convergent subsequence. We don't have boundedness here. I can see that the jumping around probably can't prevent some sequence from converging. But I think I'd have to work at a proof. Quote:
I agree with you about C and D. I must be missing something obvious. Last edited by Maschke; May 24th, 2017 at 08:00 PM.  
May 24th, 2017, 08:33 PM  #4  
Senior Member Joined: Sep 2015 From: USA Posts: 2,427 Thanks: 1314  Quote:
(A) and (B) seem fairly deep questions then.  
May 25th, 2017, 04:43 AM  #5 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,649 Thanks: 2630 Math Focus: Mainly analysis and algebra 
C and D talk about limit points of the sequence of all rationals. I'm reasonably sure that such a sequence doesn't converge and that $\limsup a_n$ and $\limsup a_n$ are $\pm \infty$. This is because we must include all arbitrarily large rationals in the sequence. The arbitrarily small ones are also there, of course, but that would just make the sequence oscillate. The fact that we need all arbitrarily small rationals leads me to suggest that there is guaranteed to be a subsequence that converges to zero. I also suspect that the set of such sequences must be uncountable. Of course all of this is intuition rather than proof. Last edited by v8archie; May 25th, 2017 at 04:46 AM. 
May 25th, 2017, 01:22 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,617 Thanks: 2072  

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rational, sequence, terms 
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