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Homework Help: Limit of Functions proof; help

  1. Nov 11, 2008 #1
    1. The problem statement, all variables and given/known data

    Let E [tex]\subset \mathbb{R}[/tex] and E [tex]\neq \emptyset[/tex]. Fix p as a limit point of E. Suppose that f is bounded and realvalued on E and that [tex]\lim_{x \to p}f(x)[/tex] does not exist. Prove the fact that there exist sequences p_n and q_n in E with [tex]\lim_{n}p_n=\lim_{n}q_n=p[/tex] such that [tex]\lim_{n}f(p_n)[/tex] and [tex]\lim_{n}f(q_n)[/tex] exist, but are different.

    2. Relevant equations

    3. The attempt at a solution

    Since p is a limit point of E, we are guaranteed a sequence (p_n) in E where [tex]p_n \neq p[/tex] and [tex]\lim_{n}p_n=p[/tex]. Using the same logic I can find another sequence, call it (q_n), that converges to p.

    Now I am unsure how [tex]\lim_{n}f(p_n) [/tex]and [tex]\lim_{n}f(q_n)[/tex] exist?
    Last edited: Nov 11, 2008
  2. jcsd
  3. Nov 11, 2008 #2


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    Staff Emeritus
    Science Advisor
    Gold Member

    f is bounded, so given a sequence of values (f(pn), you can find a convergent subsequence. Use the subsequence instead of the full sequences you originally find.

    I'd be wary about the 'using the same logic' part though... let's say E is the set of all numbers of the form 1/n... if my first sequence is an = 1/n, I've screwed myself out of finding any more sequences. You need to work your way around this problem
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