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.Originally Posted by Drexel28
Haha, true dat.
Ok, so we call $\displaystyle x$ a limit (clutser) point of $\displaystyle E$ if given any arbitrary $\displaystyle \varepsilon>0$ there exists some $\displaystyle e\in B_{\varepsilon}(x)$ such that $\displaystyle e\in E-\{x\}$.
Ok, so let $\displaystyle B_{\varepsilon}(x)$ be arbitrary.
We have two cases: either $\displaystyle \{x_n\}$ contains infinitely many points or it is eventually $\displaystyle x$. If it is the latter we may conclude. So, assume not.
Since $\displaystyle x_n\to x$ and $\displaystyle \{x_n\}$ has infinitely many points we know that given any $\displaystyle \varepsilon>0$ we have that $\displaystyle B_{\varepsilon}(x)$ contains infinitely many distinct points of $\displaystyle \{x_n\}$. In particular, there's at least two, call them $\displaystyle x_{n_1},x_{n_2}$. Since $\displaystyle x_{n_1}\ne x_{n_2}$ we cannot have that they both equal $\displaystyle x$. So, let's assume WLOG that $\displaystyle x_{n_1}\ne x$. Since $\displaystyle B_{\varepsilon}(x)$ is open we know there exists some $\displaystyle \delta>0$ such that $\displaystyle B_{\delta}(x_{n_1})\subseteq B_{\varepsilon}(x)$. But, since $\displaystyle x_{n_1}$ is a limit point of $\displaystyle E$ it must contain infinitely many distinct points of $\displaystyle E$. Namely there has to be at least two distinct points of $\displaystyle E$, call them $\displaystyle e_1,e_2$. Since they are distinct we know they both can't be equal to $\displaystyle x$. So, we may assume WLOG that $\displaystyle e_1\ne x$. But, $\displaystyle e_1\in B_{\delta}(x_{n_1})$ and since $\displaystyle B_{\delta}(x_{n_1})\subseteq B_{\varepsilon}(x)$ it follows that $\displaystyle e_1\in B_{\varepsilon}(x)$.
Since $\displaystyle \varepsilon>0$ 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 theoryEverything 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
Haha, I feel for you man. For the longest time I didn't know what W.R.T., WLOG, or N.B. meant.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
W.R.T.- With Respect To
WLOG- Without Loss Of Generality
N.B.- Nota Benne (or something...it means note well in latin I think )
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