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Math Help - Limit help

  1. #1
    Senior Member Danneedshelp's Avatar
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    Limit help

    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.
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  2. #2
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    Quote Originally Posted by Danneedshelp View Post
    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|
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  3. #3
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by Plato View Post
    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}).
    Last edited by Danneedshelp; September 19th 2009 at 08:04 PM.
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  4. #4
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    Quote Originally Posted by Danneedshelp View Post
    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
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  5. #5
    Senior Member Danneedshelp's Avatar
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    Quote Originally Posted by Plato View Post
    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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