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Math Help - Proof of Non-Uniform Continuity Criteria

  1. #1
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    Proof of Non-Uniform Continuity Criteria

    Let A\in\mathbb{R} and let f:A\rightarrow\mathbb{R}.
    How do you prove the last two statements of the criteria are equivalent?
    2 - There exists \epsilon_0 > 0 such that for every \delta>0, there exists x_{\delta},u_{\delta} \in A so that |x_{\delta} - u_{\delta}|<\delta and |f(x_{\delta}) - f(u_{\delta})| \geq \epsilon_0

    3 - There exists \epsilon_0 > 0 and there exist sequences (x_n),(u_n) in A such that u_n - x_n \rightarrow 0 as n \rightarrow \infty, yet |f(x_n)-f(u_n)| \geq \epsilon_0
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  2. #2
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    Assume #2. Then to prove #3 let \delta=\frac{1}{n} for each n\in \mathbb{Z}^+.

    Assume #3. Then for each \delta>0 some term of \left(x_n-u_n\right) must have absolute value less than \delta.
    .
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