Question

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?

Answer 1

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.

Answer 2

yes i think it is compact, which i believe means any subsequence
\[\{x_n\}_{n\geq 1}\] converges to something in the set

Answer 3

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

Answer 4

if i recall correcty ( it has been a while) sequence is compact if every subsequence contains a further subsequence that is convergent

Answer 5

That makes sense. :) Thank you

Answer 6

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

Answer 7

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