I'm trying to study for an upcoming exam for a class in which the book title is Intro to analysis, so i think it belongs in this category. However, this question is not in the book and I am completely lost. I hope someone could help so I can understand and learn the answer. Thanks!
Let E be a set of real numbers. Suppose that x = lim_{n to infinity} x_n and that each x_n is a cluster point of E. Show that x is also a cluster point of E. (Warning: the numbers x_n may not belong to E.)
Well I thought to show that there exists a sequence and that there is an e(epsilon)>0 such that |x_n-x|<e and let n>=N. But honestly I'm completely lost and confused with this question. I have a tendency to just make things up when I write proofs. So I guess i'll say I haven't tried too much.
Haha, true dat.
Ok, so we call a limit (clutser) point of if given any arbitrary there exists some such that .
Ok, so let be arbitrary.
We have two cases: either contains infinitely many points or it is eventually . If it is the latter we may conclude. So, assume not.
Since and has infinitely many points we know that given any we have that contains infinitely many distinct points of . In particular, there's at least two, call them . Since we cannot have that they both equal . So, let's assume WLOG that . Since is open we know there exists some such that . But, since is a limit point of it must contain infinitely many distinct points of . Namely there has to be at least two distinct points of , call them . Since they are distinct we know they both can't be equal to . So, we may assume WLOG that . But, and since it follows that .
Since was arbitrary the conclusion follows.
Oh man, thanks so much. Following it along while reading the def given in the book, its making sense. As you can tell I'm not fond of theory Everything else I'm good with. However, I'm not sure I'm completely following the lingo (if it is lingo), what is WLOG? Thanks again