Double Subsequences Suppose that $\{a_{n}\}$ and $\{b_{n}\}$ are bounded. Prove that $\{a_{n}b_{n}\}$ has  a convergent subsequence. 
In class this is how my professor argued:
By the Bolzano-Weierstrass Theorem, there exists a subsequence $\{a_{n_k}\}$ that converges to $a$. Since $\{b_n\}$ is bounded, $\{b_{n_k}\}$ is also bounded. So by the Bolzano-Weierstrass Theorem, there exists a subsequence of $\{b_{n_k}\}$ namely $\{b_{n_{{k_j}}}\}$ such that $\{b_{n_{{k_j}}}\}$ converges to $b$. 
In particular $\{a_{n_{{k_j}}}\}$ will converge to $a$. And note that $\{a_{n_{{k_j}}}b_{n_{{k_j}}}\}$ is a subsequence of $\{a_{n}b_{n}\}$. So $a_{n_{{k_j}}}b_{n_{{k_j}}} \to ab$. 
My question is why do we have to use so many subsequences. Is it wrong to argue as follows?
$\{a_{n}\},\{ b_{n} \}$ are both bounded, so by the Bolzano-Weierstrass Theorem, both sequences have a convergent subsequence. Namely $a_{n_k} \to a$ and $b_{n_k} \to b$. Then note that $\{a_{n_k}b_{n_k}\}$ is a subsequence of $\{a_{n}b_{n}\}$ which converges to $ab$. And we are done.

You can actually use less subsequences using a different proof, that Aryabhata is hinting at in his comment:
Proof. Sequences $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are bounded, so there exist $A\geq0$ and $B\geq0$ such that $|a_n|\leq A$ and $|b_n|\leq B$ for all $n\in\mathbb{N}$. But from this we immediately see that $|a_nb_n|\leq AB$ for all $n\in\mathbb{N}$. So $(a_nb_n)_{n=1}^\infty$ is a bounded sequence as well. Therefore by the Bolzano-Weierstrass theorem it has a convergent subsequence. $\square$
If you prove it the way your professor did, however, you have to use many sequences. To see why, I suggest you try the following. First prove:
Lemma. Suppose $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are bounded sequences and $(a_n)_{n=1}^\infty$ is convergent. Then $(a_nb_n)_{n=1}^\infty$ has a convergent subsequence.
(Hint: one of the sequences is convergent, so you don't have to worry about indices properly aligning. Can you see why?)
After you have proved this, the professor's proof will look like this:
Proof. Sequences $(a_n)_{n=1}^\infty$ and $(b_n)_{n=1}^\infty$ are bounded, so there exists a convergent subsequence $(a_{n_k})_{k=1}^\infty$ of the sequence $(a_n)_{n=1}^\infty$. This means that $(a_{n_k})_{k=1}^\infty$ and $(b_{n_k})_{k=1}^\infty$ are bounded sequences and $(a_{n_k})_{k=1}^\infty$ is convergent. So we can use the lemma on them, telling us that some subsequence of $(a_{n_k}b_{n_k})_{k=1}^\infty$ is convergent. Since a subsequence of a subsequence is again a subsequence of the original sequence, this completes the proof. $\square$
I hope this clears things up a bit.
