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.