# Thread: Convergence with products of sequences

1. ## Convergence with products of sequences

a) Let (an) be a bounded sequence (not necessarily convergent) and let (bn) be a sequence that converges to zero. Prove that the sequence of products (anbn) converges to zero.

b) Explain why you could not use the Algebraic Limit Theorem.

(My first thought was to use the ALT, so I'm confused. Not sure how to formally prove this, although it seems to make sense at the surface)

2. Originally Posted by Janu42
a) Let (an) be a bounded sequence (not necessarily convergent) and let (bn) be a sequence that converges to zero. Prove that the sequence of products (anbn) converges to zero.

b) Explain why you could not use the Algebraic Limit Theorem.

(My first thought was to use the ALT, so I'm confused. Not sure how to formally prove this, although it seems to make sense at the surface)

$\displaystyle |a_nb_n|\leq A|b_n|\,,\,\,with \,\,\,|a_n|\leq A\,\,\,\forall n\in\mathbb{N}$. Now take any $\displaystyle \epsilon > 0\Longrightarrow \exists N_\epsilon \in\mathbb{N}\,\,\,s.t.\,\,|b_n|<\frac{\epsilon}{A }\,\,\,\forall n>N_\epsilon$ ...

Tonio

3. Ok so that makes the product less than epsilon, showing it converges. How do I know that it converges to 0 though? And why is it that I can't use the ALT?

4. Originally Posted by Janu42
Ok so that makes the product less than epsilon, showing it converges. How do I know that it converges to 0 though? And why is it that I can't use the ALT?
$\displaystyle \left| {a_n b_n - 0} \right| = \left| {a_n b_n } \right| < A\frac{\varepsilon }{A}$.

I doubt that most of us even knows what ALT means.

5. Algebraic Limit Theorem, from part (B)

6. Originally Posted by Janu42
Algebraic Limit Theorem, from part (B)
That is not any help!
Why do you think that "Algebraic Limit Theorem" has any meaning to most people?

Why not state its statement?

7. Originally Posted by Janu42
Ok so that makes the product less than epsilon, showing it converges. How do I know that it converges to 0 though? And why is it that I can't use the ALT?

Uh? A sequence $\displaystyle \{c_n\}$ converges to zero iff for any $\displaystyle \epsilon > 0$ we have $\displaystyle |c_n|<\epsilon$ for all the elements of the sequence but perhaps a finite number of them, and this is exactly what I showed above for $\displaystyle \{a_nb_n\}$...

Tonio