(adsbygoogle = window.adsbygoogle || []).push({}); 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?

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# Homework Help: Limit of Functions proof; help

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