# is the limit point of $(-1)^n$ in contradiction to the definition of a limit point?

$$(-1)^n=\{-1,1,-1,1,...\}$$ Now according to the definition of limit point A point $$x$$ in $$X$$ is a limit point or cluster point or accumulation point of a set of $$S$$ if every neighborhood of $$x$$ contains at least one point of $$S$$ different from $$x$$ itself Now if I take deleted nbd of $$1$$ I don't have any element in the set. So how $$1$$ or $$-1$$ would be its limit point.

• The limit point of a sequence and the limit point of a set are two different concepts. Commented Sep 28, 2021 at 10:49
• Thank you @jjagmath Commented Sep 28, 2021 at 10:56
• I have downvoted the question because it isn't communicated well in my opinion. Commented Sep 28, 2021 at 11:21
• The equation in line 1 does not make sense. On the LHS you have a number, on the RHS a set of numbers. Commented Sep 28, 2021 at 17:35

If you were to consider $$\{-1,1,-1,...\}$$ as a set, then it would actually be $$\{-1,1\}$$ since sets do not contain the same element more than once (unless it is a multiset). In this case you are correct, this set does not have limit points.
However, if you consider the sequence $$-1,1,-1,...$$, then a limit point is one that for every neighbourhood around the point, there are infinitely many terms that lie within the neighbourhood. So $$-1$$ and $$1$$ are limit points of the sequence because they occur infinitely often.