convergence problem, help plz

decoY

Let {bn} be a bounded sequence and suppose that an -> 0 . Prove that anbn -> 0

thanks in advance

**Posted:** Fri Sep 23, 2005 5:56 am

Kent Merryfield

What does your problem really say? What you posted is so obviously false that it cannot be the statement you meant, but I'm not willing to guess what it should be.

**Posted:** Fri Sep 23, 2005 11:46 am

decoY

that is what it says
it's from the textbook: Fundamental Ideas of Analysis by Michael Reed
page 39 problem # 4.

**Posted:** Fri Sep 23, 2005 11:51 am

Kent Merryfield

Oh, sorry. I misunderstood your notation, as I was reading the symbol you meant to be $\to$ as $\ge$ , which changes the meaning.

One thing you'll find around here: knowing a little $\LaTeX$ greatly improves your ability to communicate. Here's your problem. Just wave your mouse over it or click on the coded sections to see what the code looks like.

Let $\{b_n\}$ be a bounded sequence and suppose that $a_n \to 0$ . Prove that $a_n b_n \to 0.$

Let's try walking through this. What does " $a_n\to 0$ mean? It means this:

$\forall \epsilon>0\,\exists N$ such that if $n>N,\,|a_n|<\epsilon.$

What does " $\{b_n\}$ is bounded" mean? It means this:

$\exists\,M$ such that $|b_n|\le M$ for all $n.$

Now: you want to show that $|a_nb_n-0|=|a_nb_n|<\epsilon$ for some $\epsilon.$ What are the resources you can bring to bear? What facts do you know about $a_n$ and $b_n$ that you can use?

**Posted:** Fri Sep 23, 2005 12:02 pm