Thread: Real Analysis - Limits and Order #2

Let (an) be a bounded (not necessarily convergent) sequence, and assume
lim bn = 0. Show that lim(an*bn) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this.

Hello,

Originally Posted by ajj86

Let (an) be a bounded (not necessarily convergent) sequence, and assume
lim bn = 0. Show that lim(an*bn) = 0. Why are we not allowed to use the Algebraic Limit Theorem to prove this.

What is the algebraic limit theorem ??

$\displaystyle \lim b_n=0 \Leftrightarrow \forall \epsilon > 0, ~ \exists N \in \mathbb{N}, ~ \forall n > N, ~ |b_n|< \epsilon$

$\displaystyle (a_n) \text{ bounded } \Leftrightarrow \exists K>0 \quad |a_n|<K \quad \forall n$ (in particular for n>N)

Multiply :
$\displaystyle |a_n| \times |b_n|=|a_n \times b_n|< \epsilon K$

So :
$\displaystyle \forall \delta= \epsilon K, ~ \exists N \in \mathbb{N}, ~ \forall n > N, ~ |a_n| \times |b_n|=|a_n \times b_n|< \delta$

This is the definition for $\displaystyle \lim a_n \times b_n=0$

I appreciate your answer to my question. I've been having a lot of trouble understanding real analysis. This class reminds me a lot of abstract algebra. I guess proofs just aren't my forte.