Accelerate convergence of sequence

Question

Is it always possible to extract a subsequence from my generic sequence $(q_n)$, such that the convergence of the subsequence to the same limit $r$ is faster then the original?

Answer 1

The answer is yes.

Pick any rate of convergence $r_n$ you want. Then, for each $n$ there exists some $M_n$ such that for all $m >M_n$ we have $$|a_n-l|<r_n$$

Pick inductively $$k_n > \max \{ k_1,k_2,..., k_{n-1}, M_n \}$$

Then $a_{k_n}$ is a subsequence and since $k_n >M_n$ we have $$|a_{k_n} -l| <r_n$$