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Thread: Using combination rules!!! Help please!

  1. #1
    Jan 2010

    Using combination rules!!! Help please!

    True or False, with a proof or counterexample.

    a) If bn ≠ 0 and an/bn →1, then an-bn → 0
    b) If bn ≠ 0, bn is bounded and an/bn → 1 then an-bn → 0

    At the moment I cannot even see which is false so I am struggling with this question. I think the proof will require use of the quotient combination rule and the sum combination rule but I cannot see where to start! Any help would be appreciated!
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  2. #2
    Senior Member Shanks's Avatar
    Nov 2009
    for(a), it is false. Here is a counterexaple: $\displaystyle a_n=n+1, b_n=n,\frac{a_n}{b_n}\to 1$, but $\displaystyle a_n-b_n=1\to 1$.
    for(b), It is ture! $\displaystyle \left|a_n-b_n\right|=\left|b_n\right|\cdot\left|\frac{a_n}{b _n}-1\right|\to 0$ since $\displaystyle b_n$ is bounded.
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