Results 1 to 7 of 7

Math Help - Show that x is also a cluster point of E...

  1. #1
    Member
    Joined
    Feb 2010
    Posts
    146

    Show that x is also a cluster point of E...

    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.)
    Follow Math Help Forum on Facebook and Google+

  2. #2
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    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.)
    What have you tried? Do you know what a limit point, or any of the other relevant definitions are?
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Member
    Joined
    Feb 2010
    Posts
    146
    Quote Originally Posted by Drexel28 View Post
    What have you tried? Do you know what a limit point, or any of the other relevant definitions are?
    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.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    So I guess i'll say I haven't tried too much.
    Haha, true dat.

    Ok, so we call x a limit (clutser) point of E if given any arbitrary \varepsilon>0 there exists some e\in B_{\varepsilon}(x) such that e\in E-\{x\}.

    Ok, so let B_{\varepsilon}(x) be arbitrary.

    We have two cases: either \{x_n\} contains infinitely many points or it is eventually x. If it is the latter we may conclude. So, assume not.

    Since x_n\to x and \{x_n\} has infinitely many points we know that given any \varepsilon>0 we have that B_{\varepsilon}(x) contains infinitely many distinct points of \{x_n\}. In particular, there's at least two, call them x_{n_1},x_{n_2}. Since x_{n_1}\ne x_{n_2} we cannot have that they both equal x. So, let's assume WLOG that x_{n_1}\ne x. Since B_{\varepsilon}(x) is open we know there exists some \delta>0 such that B_{\delta}(x_{n_1})\subseteq B_{\varepsilon}(x). But, since x_{n_1} is a limit point of E it must contain infinitely many distinct points of E. Namely there has to be at least two distinct points of E, call them e_1,e_2. Since they are distinct we know they both can't be equal to x. So, we may assume WLOG that e_1\ne x. But, e_1\in B_{\delta}(x_{n_1}) and since B_{\delta}(x_{n_1})\subseteq B_{\varepsilon}(x) it follows that e_1\in B_{\varepsilon}(x).

    Since \varepsilon>0 was arbitrary the conclusion follows.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Member
    Joined
    Feb 2010
    Posts
    146
    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
    Follow Math Help Forum on Facebook and Google+

  6. #6
    MHF Contributor Drexel28's Avatar
    Joined
    Nov 2009
    From
    Berkeley, California
    Posts
    4,563
    Thanks
    21
    Quote Originally Posted by tn11631 View Post
    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
    Haha, I feel for you man. For the longest time I didn't know what W.R.T., WLOG, or N.B. meant.

    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 )
    Follow Math Help Forum on Facebook and Google+

  7. #7
    MHF Contributor

    Joined
    Apr 2005
    Posts
    15,368
    Thanks
    1313
    "well" is "bene" in Latin- only one "n".
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Cluster point of a finite sequence
    Posted in the Differential Geometry Forum
    Replies: 4
    Last Post: October 22nd 2009, 08:57 AM
  2. Cluster Point
    Posted in the Differential Geometry Forum
    Replies: 3
    Last Post: August 13th 2009, 12:44 PM
  3. !Cluster Point <=> Convergent
    Posted in the Differential Geometry Forum
    Replies: 7
    Last Post: July 24th 2009, 07:53 AM
  4. Cluster point:
    Posted in the Differential Geometry Forum
    Replies: 2
    Last Post: March 16th 2009, 12:11 PM
  5. Unique Cluster Point => Convergent?
    Posted in the Calculus Forum
    Replies: 3
    Last Post: September 13th 2008, 02:21 PM

Search Tags


/mathhelpforum @mathhelpforum