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Theorems of convergence for sequences?

  1. Oct 18, 2005 #1
    For a sequence a_1, a_2, ... in R^n to be convergent there are (at least) 2 theorems, as follows:

    if for all epsilon>0 there exists an M such that when m>M, then |a_m-a|<epsilon

    and also:
    If u(epsilon) is a function such that u(epsilon)-->0 as epsilon-->0, then
    the sequence is convergent if

    for all epsilon>0 there exists an M such that when m>M, then |a_m-a|<epsilon

    ~~~~~~~~~~~~
    I can understand this intuitively, because u(epsilon) and epsilon behave similarly .. but how I prove that they are equivalent? Do I simply take the limit of both definitions, and then set them equal? help!
     
  2. jcsd
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