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Difficulty with accumulations points

  1. Oct 27, 2012 #1
    1. The problem statement, all variables and given/known data
    Hi guys,

    I'm having real difficulty with understanding accumulation points. I don' really know why that is since others seem to understand the concept fine but I'm very lost.
    For example, I'm practicing some questions and one of the is :
    If S is the set of rational numbers with 1<x<2, then is √2 is an accumulation point?

    I am completely lost in how to go about figuring this out.
    Examples help, so if you have any good examples that could make this concept a little clearer, I would truly appreciate it.

    Thanks!

    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 27, 2012 #2

    HallsofIvy

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    What, exactly, is your understanding of the definition of "accumulation point"? One that is commonly used is "p is an accumulation point of set A if and only if there exist a sequence of points in A (not including p) that converges to p. Another is that every neighborhood of p contains at least one point of A (other than p).

    [itex]\sqrt{2}= 1.41421...[/itex], right? So given any [itex]\delta> 0[/itex], there exist a power of 10 such that [itex]10^n< \delta[/itex]. Cutting that number off after n decimal places gives a rational number closer to [itex]\sqrt{2}[/itex] than [itex]\delta[/itex].

    More generally, given any real number, there exist a sequence of rational numbers converging to it.
     
  4. Oct 27, 2012 #3
    I struggle with how/what process I need to follow to find accumulation pts of a set or determining of a given value is an accumulation pt.
     
  5. Oct 27, 2012 #4

    HallsofIvy

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    Do you know what the definition of "accumulation point" is? It sounds like you are saying you don't.
     
  6. Oct 27, 2012 #5
    You are right, I have the definition but that's when I'm struggling...truly understanding it and applying it.
     
  7. Oct 27, 2012 #6

    haruspex

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    Does it help to put it like this: p is an accumulation point of S if you can get arbitrarily close to it by picking points of S-{p}.
    There's any number of sequences of rationals in [1,2] that converge to sqrt(2). Halls gave you a very easy and obvious one. Another is to start with x = 1 and generate a sequence of rationals by iterating x' = 1/(1+x) + 1. (You can get that formula by writing y2 - 1 = 1, so (y-1) = 1/(y+1).)
     
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