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Subtracting radical functions

  1. Aug 13, 2013 #1
    This is something that comes up when I want to determine whether the sequence of functions {f_n} converge uniformly to f:

    Suppose f_n(x) = sqrt(x^2 + 1/n^2), so f(x) = x.
    Then, according to Spivak, f(x) - f_n(x) = sqrt(x^2) - sqrt(x^2 + 1/n^2) = 1/(2n^2*sqrt(ε)) for some ε such that x^2 < ε < x^2 + 1/n^2.

    Similarly, sqrt(x) - sqrt(x + 1/n) = 1/(2n sqrt(ε)) for some ε such that x < ε < x + 1/n.

    Why is this?

    I'd really appreciate any help. Thanks!
  2. jcsd
  3. Aug 13, 2013 #2
    Never mind--figured it out.
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