ChrisS
  • ChrisS
Could someone explain to me what an accumulation point is in fairly plain language? I have a definition here but it's not really clicking for me.
Mathematics
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chestercat
  • chestercat
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amistre64
  • amistre64
whats your definition?
ChrisS
  • ChrisS
Let S be a set of real numbers. A real number A is an accumulation point of S iff every neighborhood of A contains infinitely points of S.
amistre64
  • amistre64
sounds to me almost like a limit everything is accumulating in one area; or amassing at a concentrated point

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Zarkon
  • Zarkon
another name for accumulation point is limit point ;)
amistre64
  • amistre64
:) cool
amistre64
  • amistre64
http://mathworld.wolfram.com/AccumulationPoint.html
ChrisS
  • ChrisS
Right, it's to do with limits. This section is on Cauchy Sequences and leads into limits.
ChrisS
  • ChrisS
so is it a point that a given sequence will converge to?
ChrisS
  • ChrisS
I guess it's one of them because Bolzano-Weierstrass implies that there can be more than one.
Zarkon
  • Zarkon
you just need to make sure that the have some \(\epsilon\)-neighborhood about the point that still touches the the set S
Zarkon
  • Zarkon
Isolated points can't be limit points.
Zarkon
  • Zarkon
so not every sequence leads to a limit point [I don't like the term accumulation point ;)]
ChrisS
  • ChrisS
but every convergent (or Cauchy) sequence will have one right?
ChrisS
  • ChrisS
because all Cauchy sequences are bounded, and all bounded sequence has at least one limit point
ChrisS
  • ChrisS
*sequences have...
Zarkon
  • Zarkon
what if \[S=\{0\}\cup[1,2]\] let \[a_n=0,\,\,\, \forall n\in\mathbb{N}\]
Zarkon
  • Zarkon
then \(\{a_n\}\) is a Cauchy sequence
Zarkon
  • Zarkon
but \(\{0\}\) is an isolated point and therefore not a limit point
ChrisS
  • ChrisS
fire alarm at the school :/ I'll be back as soon as I can.
ChrisS
  • ChrisS
Back. Thanks for the help. I was able to catch a professor outside while we were waiting and I think I have it down now.
Zarkon
  • Zarkon
good to hear

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