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Math Help - Proof - limit of a sequence

  1. #1
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    Proof - limit of a sequence

    Prove, by definition, that if \lim_{n\to \infty} a_n = L then \lim_{n\to \infty} a_n^2 = L^2.

    It should be a basic proof but I'm somehow stuck. I was obviously thinking:

    \mid a_n^2 - L^2 \mid = \mid a_n - L \mid \cdot \mid a_n + L\mid

    but I'm not sure how to proceed.
    Any help would be appreciated.
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  2. #2
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    Re: Proof - limit of a sequence

    |a_n-L| is small and |a_n+L| is close to 2L and therefore cannot be large.
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  3. #3
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    Re: Proof - limit of a sequence

    I don't understand...
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  4. #4
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    Re: Proof - limit of a sequence

    You are given an ε > 0 and you need to make |aₙ - L|⋅|aₙ + L| < ε by choosing n. You can make |aₙ - L| arbitrarily small by choosing n big enough. Also, you can make |aₙ + L| arbitrarily close to 2L by choosing n big enough. In particular, you can make |aₙ + L| < 3L for some n and onward. Assuming you have done this, how small do you need to make |aₙ - L| so that |aₙ - L|⋅|aₙ + L| becomes smaller than ε?
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