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Math Help - Pretty confusing proof: I don't even know what it is I am supposed to prove...

  1. #1
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    Pretty confusing proof: I don't even know what it is I am supposed to prove...

    Let f-->R and xo is a cluster point of D. Prove that f has a limit at xo if for each epsilon>0 there is a neighborhood Q of xo such that for all x,y belonging to QintersectD x=/=xo, y=/=xo, we have |f(x)-f(y)|<epsilon.

    I have an exam coming up in a week and I don't even know where to start with such weird problems.
    I found a proof of this statement on the web (Since we don't have a solution manual), but it is very long and complicated..using many things that we have not used in this course. I was hoping someone could help me get an easier proof so that if such a problem appeared on my exam, I might actually have a chance because there is no way I could come up with the proof I found. Thanks
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by zhupolongjoe View Post
    Let f-->R and xo is a cluster point of D. Prove that f has a limit at xo if for each epsilon>0 there is a neighborhood Q of xo such that for all x,y belonging to QintersectD x=/=xo, y=/=xo, we have |f(x)-f(y)|<epsilon.

    I have an exam coming up in a week and I don't even know where to start with such weird problems.
    I found a proof of this statement on the web (Since we don't have a solution manual), but it is very long and complicated..using many things that we have not used in this course. I was hoping someone could help me get an easier proof so that if such a problem appeared on my exam, I might actually have a chance because there is no way I could come up with the proof I found. Thanks
    Couple of clarification questions:

    What do you mean by:

    f-->R

    (Last I checked, "Big Grin" was not a set...)

    And what set are you denoting as D?
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  3. #3
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    Oops lol, didn't notice that. It's just any set D
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  4. #4
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by zhupolongjoe View Post
    Let f: D-->R and xo is a cluster point of D. Prove that f has a limit at xo if for each epsilon>0 there is a neighborhood Q of xo such that for all x,y belonging to QintersectD x=/=xo, y=/=xo, we have |f(x)-f(y)|<epsilon.

    I have an exam coming up in a week and I don't even know where to start with such weird problems.
    I found a proof of this statement on the web (Since we don't have a solution manual), but it is very long and complicated..using many things that we have not used in this course. I was hoping someone could help me get an easier proof so that if such a problem appeared on my exam, I might actually have a chance because there is no way I could come up with the proof I found. Thanks
    Let me clean this up a bit:

    Let f: D\longrightarrow\mathbb{R} be a function and let x_0 be a cluster point of D. Prove that \lim_{x_n\to x_0}f(x_n) exists if \forall~\epsilon>0, \exists~\delta such that x,y\in Q_{\delta}(x_0) implies that |f(x)-f(y)|<\epsilon.

    ----------

    Proof: Let \{x_n\}\subset D be a sequence that converges to x_0. (We know such a sequence exists because x_0 is a cluster point.) We want to prove that the sequence \{f(x_n)\}\subset\mathbb{R} converges.

    We know that given a \delta, all but finitely many x_n are contained in Q_{\delta}(x_0), that is, n>N implies x_n\in Q_{\delta}(x_0). So for n,m>N we know that x_m,x_n\in Q_{\delta}(x_0) and therefore |f(x_n)-f(x_m)|<\epsilon.

    But this means that \{f(x_n)\} is Cauchy, and therefore converges because \mathbb{R} is complete. In other words, \lim_{x_n\to x_0}f(x_n) exists. \square
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  5. #5
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    Thanks a bunch! I understood that
    a lot more than the other proof I found. BTW: I love your David Hilbert quote...
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