Limit of Functions proof; help

[Unassuming]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

Now I am unsure how $$\lim_{n}f(p_n)$$and $$\lim_{n}f(q_n)$$ exist?

 

[Office_Shredder]

f is bounded, so given a sequence of values (f(p_n), 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 a_n = 1/n, I've screwed myself out of finding any more sequences. You need to work your way around this problem