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?
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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 $$|a_{k_n} -l| <r_n$$
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$\begingroup$ So we are basically choosing the natural number $M_n$ so that we make $\epsilon$ small enough (what you called $r_n$) to convergence, at every step, at the rate I want? $\endgroup$ Commented Jan 15, 2021 at 17:28
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1$\begingroup$ @TheTurtleHermit For each $n$ you set $\epsilon$ to be the rate of convergence at that $n$ and pick one element which satisfies that inequality, yes. $\endgroup$– N. S.Commented Jan 15, 2021 at 17:37