# A question on sequences

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 $\infty$?

• First of all, as $(A_n)$ and $(B_n)$ don't relate in any way, it is sufficent, that each bounded sequence has a convergent subsequence. Therefore see en.wikipedia.org/wiki/Bolzano%E2%80%93Weierstrass_theorem – k1next Dec 23 '12 at 7:10
• So what does that question mean? I'm puzzled... – Harold Dec 23 '12 at 7:28

1. Apply the first result to the sequence $A_1,A_2,\ldots$ to get a converging subsequence $A_{n_1},A_{n_2},\ldots$
2. Apply the first result to the sequence $B_{n_1},B_{n_2},\ldots$ to get - hmm, you think about this a bit
3. Apply the second result to the sequence $A_{n_{k_1}},A_{n_{k_2}},\ldots$
• Ah Sorry. I thought it was a typo (I thought you meant $B_1,B_2,...$). I am removing my comment – Nameless Dec 23 '12 at 7:41