Limit help

**Danneedshelp** · September 19th 2009, 02:42 PM

Q: Let $(a_{n})$ be a bounded (not necessarily convergent) sequence, and assume $lim$ $b_{n}=0$ . Show that $lim(b_{n}a_{n})=0$ . 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.

**Plato** · September 19th 2009, 02:52 PM

Originally Posted by Danneedshelp

Q: Let $(a_{n})$ be a bounded (not necessarily convergent) sequence, and assume $lim$ $b_{n}=0$ . Show that $lim(b_{n}a_{n})=0$ .

Suppose that $B > 0\;\& \;\left( {\forall n} \right)\left[ {\left| {a_n } \right| \leqslant B} \right]$ .
Now in definition of $\left( {b_n } \right) \to 0$ use $\frac{\varepsilon }{B} > 0$ .
Notice that $\left| {a_n b_n - 0} \right| = \left| {a_n } \right|\left| {b_n } \right|$

**Danneedshelp** · September 19th 2009, 03:14 PM

Originally Posted by Plato

Suppose that $B > 0\;\& \;\left( {\forall n} \right)\left[ {\left| {a_n } \right| \leqslant B} \right]$ .
Now in definition of $\left( {b_n } \right) \to 0$ use $\frac{\varepsilon }{B} > 0$ .
Notice that $\left| {a_n b_n - 0} \right| = \left| {a_n } \right|\left| {b_n } \right|$

So, can I use that fact that $|a_{n}|\leq\\B$ to write: $|B||b_{n}|$ $<B\frac{\epsilon}{B}=\epsilon$ ? Since $B$ is kind of a worst case estimate of $(a_{n})$ .

**Plato** · September 19th 2009, 03:18 PM

Originally Posted by Danneedshelp

So, can I use that fact that $|b_{n}|\leq\\B$ to write: $|B||a_{n}|$ $<B\frac{\epsilon}{B}=\epsilon$ ? Since $B$ is kind of a worst case estimate of $(a_{n})$ .

You have everything backwards!
$\color{red}|a_{n}|\leq\\B$

**Danneedshelp** · September 19th 2009, 08:01 PM

Originally Posted by Plato

You have everything backwards!
$\color{red}|a_{n}|\leq\\B$

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.

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