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Prove 1/4 is not a limit point of 1/n where n is a positive integer.

Prove 1/4 is not a limit point of 1/n where n is a positive integer.
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Answer #1

For a limit point x of a set S, every neighborhood of x contains infinitely many points of S.

In your example, S = {1/n ; n belongs to N } = {1, 1/2, 1/3, 1/4, ...................}

we choose ϵ = 0.01 > 0, so ϵ neighborhood of 1/4 contains no point of S other than 1/4, i.e.

((1/4) - 0.01 , (1/4)+ 0.01) ∩ S = (0.24, 0.26) ∩ S = 1/4 (0.24) , isolated point.

every neighborhood of 1/4 does not contain infinitely many points of the set S. Hence 1/4 is not a limit point of S.

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