Question

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.

Answer 1

whats your definition?

Answer 2

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.

Answer 3

sounds to me almost like a limit

everything is accumulating in one area; or amassing at a concentrated point

Answer 4

another name for accumulation point is limit point ;)

Answer 5

:) cool

Answer 6

http://mathworld.wolfram.com/AccumulationPoint.html

Answer 7

Right, it's to do with limits. This section is on Cauchy Sequences and leads into limits.

Answer 8

so is it a point that a given sequence will converge to?

Answer 9

I guess it's one of them because Bolzano-Weierstrass implies that there can be more than one.

Answer 10

you just need to make sure that the have some \(\epsilon\)-neighborhood about the point that still touches the the set S

Answer 11

Isolated points can't be limit points.

Answer 12

so not every sequence leads to a limit point [I don't like the term accumulation point ;)]

Answer 13

but every convergent (or Cauchy) sequence will have one right?

Answer 14

because all Cauchy sequences are bounded, and all bounded sequence has at least one limit point

Answer 15

*sequences have...

Answer 16

what if \[S=\{0\}\cup[1,2]\]

let \[a_n=0,\,\,\, \forall n\in\mathbb{N}\]

Answer 17

then \(\{a_n\}\) is a Cauchy sequence

Answer 18

but \(\{0\}\) is an isolated point and therefore not a limit point

Answer 19

fire alarm at the school :/ I'll be back as soon as I can.

Answer 20

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.

Answer 21

good to hear