1. Not finding help here? Sign up for a free 30min tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Lim a(n)b(n) = AB

  1. Feb 19, 2014 #1
    1. The problem statement, all variables and given/known data

    Theorem. If lim a n = A and lim b n = B, then lim a n b n = AB.




    2. Relevant equations

    |an - A| < ε

    |bn - B| > ε



    3. The attempt at a solution

    I have the solution to this, but I'm unclear on why one part is done.

    Let ε > 0. Since bn converges it is bounded and there exists an M1 such that for all n [itex]\in[/itex] N we have |bn| < M1. Define M1 [itex]\geq[/itex] 1. Then |bn| / M1 ≤ 1.

    Skipping ahead, do the same for an and let |an - A| < [itex]\frac{ε}{2M1}[/itex]

    and let

    |bn - B| < [itex]\frac{ε}{2M2}[/itex].


    |anbn - AB| = |anbn -Abn + Abn - AB|

    So are we trying to manipulate the limit anbn into the familiar definition of |an - L| < ε so we can compare them?

    So |anbn -Abn + Abn - AB|



    |anbn -Abn| + |Abn - AB|

    =

    |an - A||bn| + |bn - B||A|

    <

    (ε/2M1)|bn|

    +

    (ε/@M2)|A|

    Stupid question, but how are we multiplying each ε by |bn| and |A|? I know that it gives us the two original lim an and lim bn, but how can we just stick them on the other side of the "<" symbol? And how do |bn|/M1 = 1 and |A|/M2 = 1 if M is just defined as being larger than both? Can we just take the max of |bn| and |A|?
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Feb 19, 2014 #2

    SammyS

    User Avatar
    Staff Emeritus
    Science Advisor
    Homework Helper
    Gold Member

    (In LaTeX, don't use the special characters at the right, like "ε" . Use \varepsilon . For subscripts use M_1, etc.)

    First it will help to state what is required to show that ##\displaystyle \lim\ \left(a_n\ b_n\right) = AB \ ## .

    You know that the sequences an and Bn converge, so you can use any positive quantity to represent "ε" for each of these sequences.
     
  4. Feb 20, 2014 #3
    |an - A||bn| + |bn - B||A|

    You know that bn converges thus there is a M such that bn<M for all n. Since |an-A| can be made really small there is a n such that for all n> N there holds

    |an - A| < epsilon/(2*(M+1)).

    but then for all n>N:

    |an - A||bn| < epsilon*M/(2*(M+1)) < epsilon /2.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Lim a(n)b(n) = AB
  1. Lim (n!)^1/n (Replies: 13)

Loading...