Let $(A_n), (B_n)$ be two bounded sequences.
Show that there is a sequence of natural numbers $n_1 < n_2 <\cdots$ so that both the subsequences $(A_{n_k})$ and $(B_{n_k})$ converge.
My problem with solving this: Is it possible to say that assuming $A_n > B_n$ for all $n$ then we can make a subsequence of $n_1 < n_2 <\cdots$ from $B_n$ to $\infty$ and then from $A_n$ to $\infty$?
Thanks in advance!