1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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:

    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

    \begin{array}{cc}\lim f\left( x_{n} \right) = A\\\lim f\left( x_{n} \right) = B\end{array}\right

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

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

    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


    User Avatar
    Science Advisor

    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


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member
    Dearly Missed

    Why can't you construct the sequence:
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook