I was thinking about the following problem:
Consider a sequence $a_n \to 0, a_n >0$ which is monotone, i. e. $a_n\ge a_{n+1}$.
Now suppose for every $C>0$ there is a subsequence $n(k)$ such that $a_{n(k)}>C\ f(n(k))$, where f is a function fulfilling $f(x)\to 0 \ (x\to\infty)(e.g. f(x)=x^{-1})$.
Can you conclude now $a_{n}>C'\ f(n)$ for all $n$?
I think so, because each subsequence $a_{n(1)-i},a_{n(2)-i},\ldots$ can not decay faster than $f$. Even when the difference $|n(k)-n(k-1)|$ is not bounded, you can choose subsequences
$a_{n(1)-1},a_{n(2)-1},\ldots$
$a_{n(1)-2},a_{n(2)-2},\ldots$
$\ldots$
$a_{1},a_{n(2)-n(1)+1},\ldots$
$a_{n(2)-n(1)},a_{n(3)-n(1)},\ldots$
and so on. Again each of these subsequence can not decay faster than $f$.
Am I doing any mistake?
Thanks for your help!