Proving Limit Points in Point Sets for Scientists

[Timberwoods]

i just ran into a hard problem, may be any of you guy can help...
prove that if the point p is a limit point of H U K where H and K are point sets, then p is a limit point of H or p is a limit point of K.
Given definition of a limit point is: a point p is said to be a limit point of a point set M if every region containing p contains a point of M distinct from p.
Thanks for your time.

 

[matt grime]

So there is a sequence in HuK tending to p. It must have a subsequnce lying in either H or K, musn't it?

 

[HallsofIvy]

Not really a "set" problem is it? I assume you have a topology on a set having H and K as subsets and a "region" is an open set in that topology.

Suppose p is a limit point of HUK. Then, by definition, each region of p (every open set containing p) contains a point of HUK different from p.

If p is a limit point of H, we are done so we can assume that is not true.
(This is a standard technique: we are asked to prove "a OR b" so we assume a is NOT true and prove b must be true.)

If p is NOT a limit point of H, there exist a region V containing p which contains no member of H (other than, possibly, p itself). Of course, since p is a limit point of HUK, V must contain a member of HUK which means it must contain a member of K. Let U be any region containing p and look at U intersect V (which is non-empty).

 

[Timberwoods]

thanks, it helps a lot, but in doing so, i still need to prove that the intersection of 2 regions is a region, which i haven't proved it yet, would you give me a hand on that?