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Math Help - Questions on cauchy definition of limit

  1. #1
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    Questions on cauchy definition of limit

    I don't understand this proof. Can you clarify this for me?

    Suppose {an}converges, then the limit is unique.

    pf) Suppose {an} converges to \alpha, \beta at the same time.

    Take any epsilon > 0,

    there exists N1  \in N such that  \mid a_n- \alpha \mid \leq epsilon/2

    there exists N2  \in N such that  \mid a_n- \beta \mid \leq epsilon/2

    Suppose N = Max(N1 , N2). <---- I don't understand this part. Why do you take the maximum of N1 and N2?

     \mid \alpha - \beta \mid = \mid \alpha - a_n+ a_n - \beta \mid \leq \mid \alpha - a_n\mid + \mid a_n- \beta \mid \leq \epsilon

    Therefore, alpha = beta
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  2. #2
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    Re: Questions on cauchy definition of limit

    If you take the smaller of the two N_i, one of your inequalities |a_n-\alpha|\leq \frac{\epsilon}{2} or |a_n-\beta|\leq \frac{\epsilon}{2} might not hold, since this is only guaranteed for n\geq N_1 and n\geq N_2.
    Thanks from rokman54
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  3. #3
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    Re: Questions on cauchy definition of limit

    The definition of convergence involves "if n> N". By taking N equal to the larger of N_1 and N_2, " n> N" gives both " n> N_1" and " n> N_2".
    Thanks from rokman54
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