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Math Help - Having trouble with understanding continuity proof

  1. #1
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    Having trouble with understanding continuity proof

    I'm having trouble with understanding one detail in this proof. I'd greatly appreciate if anyone can help.
    Let f:[0,\infty\) \rightarrow R be a continuous function. Suppose \lim_{x \to \infty} f(x)= L exists and finite. Then f is uniformly continuous on [0,\infty\).
    Proof: Let \epsilon >0. Since f(x) \rightarrow L as x \rightarrow \infty, there exists N>0 st when x>N we have \mid f(x)-L \mid <\frac{\epsilon}{2}. We know f is uniformly continuous on [0,2N] since [0,2N] is compact and f is continuous. Hence, there exists \delta_1>0 st for every x,y \in [0,2N] , \mid x-y \mid < \delta_1 \Rightarrow \mid f(x)-f(y)\mid < \epsilon. Choose \delta=min\{\delta_1,N/2\}. Then any two points x,y \in [0,\infty) satisfying \mid x-y \mid <\delta are either both in \[0,2N] or both in [N,\infty). I understand the rest of the proof. The important detail that I don't get is this line
    Choose \delta=min\{\delta_1,N/2\}. Then any two points x,y \in [0,\infty) satisfying \mid x-y \mid <\delta are either both in \[0,2N] or both in [N,\infty)
    I don't see why x,y must be both in [0,2N] or both in [N,\infty). I guess it must have something to do with the distance between them less than delta, but I don't see it. I also don't see why we need the distance less than N/2.
    Last edited by jackie; February 17th 2013 at 12:48 AM.
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  2. #2
    Senior Member jakncoke's Avatar
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    Re: Having trouble with understanding continuity proof

    if both points are not both either in [0,2N] or [N,  \infty ] this means one point has to be in [0,N) and the other point in (2N,  \infty ), meaning that their distance is greater than N. Now since  \delta is at least  \frac{N/2} (because of using min( \{\delta, \frac{N}{2}\}), this cannot be so both points have to be in [0,2N] or [N, \infty]
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