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.
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,
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 $\displaystyle \lim a_n \times b_n=0$