## anonymous 4 years ago 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?

1. 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.

2. anonymous

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

3. 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

4. anonymous

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

5. anonymous

That makes sense. :) Thank you

6. 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

7. 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