I am studying Bolzano Weierstrass theorem. It states that if the set is bounded with infinitely many points, then it has an accumulation point. Can someone please give an example of a bounded set with infinitely many points? I understand what accumulation point is, but I can't visualize this.

Thanks in advance!


Example 1: $[0,1]$.

In this bounded set, every point is an accumulation point.

Example 2: $\{1,1/2,1/3,\ldots,1/n,\ldots\}$.

In this bounded set, no point in the set is an accumulation point, but still $0$ is an accumulation point that does not lie in the set.

Example 3: $\{1,1/2,1/3,\ldots,1/n,\ldots\} \cup \{0\}$.

Starting with Example 2 we get throw in the accumulation point to get a closed set.

  • $\begingroup$ Thanks for the answer. Can you please tell why in the second example, you say that no point in the set is an accumulation point? Could you please justify that in the face of the definition? I am new to this stuff, and i haven't digested it much $\endgroup$ – Marion Crane Nov 6 '14 at 2:55
  • $\begingroup$ I say that no point in the second set is an accumulation point because for every point $P$ in the set, there is an open interval $(P-\delta,P+\delta)$ which contains no element of the set other than $P$. If $P = \frac{1}{n}$ we can take $\delta = \frac{1}{n} - \frac{1}{n+1}$. $\endgroup$ – Pete L. Clark Nov 6 '14 at 3:18

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