I've been studying about sequences and I started to read about the Bolzano–Weierstrass theorem as stated:
"Every bounded sequence {a$_n$} has a convergent subsequence"
So I have a question, if I have the sequence: ${a_n}=(-1)^n $, that is, ${a_n}=-1, 1, -1, 1, …$
It is clearly bounded within the interval: $[-1, 1]$
Since a subsequence is monotone. How can I find a subsequence for it if a term is greater than the previous one, and then the following is lower than the previous one?
Best Answer
Note that for a sub-sequence to be valid, its indexes must be increasing - not the values. So, the sequence of indexes $1,3,5$ is valid, but $1,1,3$ or $1,3,2$ aren't.
So, a converging sub-sequence to your original one will be $$1,1,1,1,\ldots \to 1$$ or $$-1,-1,-1,-1, \ldots \to -1$$