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I have the following two sets:

$A = \{{\frac{n}{n+1},\ n = 1,2,3..}\}$

 

$B = \{\frac{-n+2}{n+1},\ n = 0,1,2..\}$

And I need to find the limit points of $A$ and $B$. For each, I know that if I can find a convergent subsequence that converges to a value in these sets that is not included in their respective set definitions. Is this the proper thought process?

For $A$, the set converges to 1, despite 1 not being in the set.

For $B$, the set converges to -1, despite 1 not being in the set.

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    $\begingroup$ I think you mean $-1$ for $B$. $\endgroup$ – James Nov 19 '15 at 1:05
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    $\begingroup$ Also, I feel I should point out that neither $A$ nor $B$ are sequences, so, there is no notion of sub-sequence, nor of the set "converging". A limit point of a set $X$ in some ambient space $S$ is some $x\in S$ so that there is a sequence $\{x_n\}_{n=1}^\infty$ converging to $x$, and each $x_n\in X$. Edit: depending on your definitions, it may also be required that the sequence is never equal to the limit. $\endgroup$ – James Nov 19 '15 at 1:09
  • $\begingroup$ In this case you're mostly correct. However it's not necessary that the limit point is not included in the set, for example consider $A=\{0,1,\frac 12, \frac 13,\frac 14,... \}$. The point $0$ is a limit point of $A$ while being in the set. $\endgroup$ – BigbearZzz Nov 19 '15 at 1:09
  • $\begingroup$ @James Yes, thank you. $\endgroup$ – John Yates Nov 19 '15 at 1:10
  • $\begingroup$ @BigbearZzz, then how should I change my understanding of the limit point? Is it any convergence value of a convergent subsequene? $\endgroup$ – John Yates Nov 19 '15 at 1:11

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