1. ## Limit help

Q: Let $\displaystyle (a_{n})$ be a bounded (not necessarily convergent) sequence, and assume $\displaystyle lim$ $\displaystyle b_{n}=0$. Show that $\displaystyle lim(b_{n}a_{n})=0$. Why are we not allowed to use the Algebraic Limit Theorem to prove this?

For the last sentence is seems clear that we cannot use the ALT to prove this, because we need both limits to converge to use the ALT.

For the proof I have written out the definitions that pertain to statement; such as, the definition of a bounded sequence and convergence. I have also drawn a picture and I think I understand why this statement is true, but I am having trouble formalizing a proof.

Any help would be great.

2. Originally Posted by Danneedshelp
Q: Let $\displaystyle (a_{n})$ be a bounded (not necessarily convergent) sequence, and assume $\displaystyle lim$ $\displaystyle b_{n}=0$. Show that $\displaystyle lim(b_{n}a_{n})=0$.
Suppose that $\displaystyle B > 0\;\& \;\left( {\forall n} \right)\left[ {\left| {a_n } \right| \leqslant B} \right]$.
Now in definition of $\displaystyle \left( {b_n } \right) \to 0$ use $\displaystyle \frac{\varepsilon }{B} > 0$.
Notice that $\displaystyle \left| {a_n b_n - 0} \right| = \left| {a_n } \right|\left| {b_n } \right|$

3. Originally Posted by Plato
Suppose that $\displaystyle B > 0\;\& \;\left( {\forall n} \right)\left[ {\left| {a_n } \right| \leqslant B} \right]$.
Now in definition of $\displaystyle \left( {b_n } \right) \to 0$ use $\displaystyle \frac{\varepsilon }{B} > 0$.
Notice that $\displaystyle \left| {a_n b_n - 0} \right| = \left| {a_n } \right|\left| {b_n } \right|$
So, can I use that fact that $\displaystyle |a_{n}|\leq\\B$ to write: $\displaystyle |B||b_{n}|$$\displaystyle <B\frac{\epsilon}{B}=\epsilon? Since \displaystyle B is kind of a worst case estimate of \displaystyle (a_{n}). 4. Originally Posted by Danneedshelp So, can I use that fact that \displaystyle |b_{n}|\leq\\B to write: \displaystyle |B||a_{n}|$$\displaystyle <B\frac{\epsilon}{B}=\epsilon$? Since $\displaystyle B$ is kind of a worst case estimate of $\displaystyle (a_{n})$.
You have everything backwards!
$\displaystyle \color{red}|a_{n}|\leq\\B$

5. Originally Posted by Plato
You have everything backwards!
$\displaystyle \color{red}|a_{n}|\leq\\B$

Opps! My mistake. I just did a sloppy copy and paste job. Thats embarrassing haha. Sorry about that.

Thanks a lot for the help. I really appreciate it.