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Please explain use of Heine in proof of simple theorem

  1. Jan 27, 2005 #1
    Hi all,

    I don't fully understand the proof of uniqueness of limit of function. Our teacher proved it using the Heine's theorem. Here it is:

    Proof:
    Let [itex]\lim_{x \rightarrow a} f(x) = A[/itex] and [itex]\lim_{x \rightarrow a} f(x) = B[/itex].

    Let [itex]\left{ x_{n} \right}[/itex] satisfies: [itex]\lim_{n \rightarrow \infty} x_{n} = a[/itex]. Then


    [tex]
    \Longrightarrow^{Heine}
    \begin{array}{cc}\lim f\left( x_{n} \right) = A\\\lim f\left( x_{n} \right) = B\end{array}\right
    [/tex]

    Ok, I understand this, because according to Heine it's equivalent. But I don't get the next step:

    [tex]
    \Longrightarrow^{\mbox{Uniqueness of limit of sequence}} A = B \Longrightarrow^{Heine 2} \mbox{Uniqueness of limit of function}
    [/tex]

    How can I use the uniqueness of limit of sequence here, when [itex]f\left( x_{n} \right)[/itex] is not a sequence, but a function?

    Thank you for the explanation.
     
  2. jcsd
  3. Jan 27, 2005 #2

    HallsofIvy

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    If lim(x->a) f(x)= A, then f(xn) MUST also converge to A for any sequence xn converging to a. If f converged to both A and B, then the sequence f(xn) must converge to both A and B which, apparently, you have already proved is impossible.

    Actually the proof of the uniqueness the limit of a sequence can be modified to give directly a proof of the uniqueness of the limit of a function.

    If f converges to both A and B, take ε to be half the distance from A to B. Then show that x can't be within &epsilon of both A and B.
     
  4. Jan 27, 2005 #3
    Well, that's what I don't understand. I think that [itex]f\left(x_{n}\right)[/itex] is function, not sequence. Sequence goes only over integers, whereas [itex]f\left(x_{n}\right)[/itex] doesn't...That's why I can't use the uniqueness of limit of sequence directly here I think...Of course there are other ways how to prove the uniqueness of limit of function, I just want to understand this one.
     
  5. Jan 27, 2005 #4

    arildno

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    Why can't you construct the sequence:
    [tex]a_{n}=f(x_{n})[/tex]??
     
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