anonymous
  • anonymous
How can I tell if the sequence\[\left(1,\frac{1}{2},\frac{2}{3},\frac{3}{4},\frac{4}{5},\frac{5}{6},...\right)\]is compact or not?
Mathematics
  • Stacey Warren - Expert brainly.com
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SOLVED
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jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
I noticed that the sequence converges to \(1\), which is in the set. I also noticed that every subsequence will also converge to \(1\). To me, it seems the sequence is compact.
anonymous
  • anonymous
yes i think it is compact, which i believe means any subsequence \[\{x_n\}_{n\geq 1}\] converges to something in the set
anonymous
  • anonymous
of course it is always possible that your subsequence only has a finite number of distinct elements, but that is ok because if such is the case then at least one must occur infinitely often and you can define the subsequence \[\{x_{n_k}\}=x\] for such an x

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anonymous
  • anonymous
if i recall correcty ( it has been a while) sequence is compact if every subsequence contains a further subsequence that is convergent
anonymous
  • anonymous
That makes sense. :) Thank you
anonymous
  • anonymous
yw but you still have to say something. for example if you have a subsequence \[\{x_n\}\] you have to explain why it would have a further subsequence \[\{x_{n_k}\}\] that converges
anonymous
  • anonymous
so for example you could make \[\{x_{n_k}\}\] be something like \[x_k \in \{x_n\} \cap (1-\frac{1}{k},1+\frac{1}{k})\]man that took a long time to type

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