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limit points of set, closure of set, isolated points of set

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Let B = {[(-1^n](n)/(n + 1): n = 1, 2, 3, ...}.

(a) Find the limit points of B.

(b) Is B a closed set?

(c) Is B an open set?

(d) Does B contain any isolated points?

(d) Find the closure of B.

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The limit points of B and the closure of B were found. Also, it was determined whether B is open, whether B is closed, and whether B contains any isolated points. A detailed explanation was given for each part of the solution.

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B = {[(-1)^n](n)/(n+1): n = 1, 2, 3, ...}

(a) Find the limit points of B.

Letting n = 1, 2, 3, ... in succession, we find that B consists of the points -1/2, 2/3, -3/4, 4/5, -5/6, 6/7, ...

The sequence of positive points (which consists of all the points for even values of n) converges to +1, and the sequence of negative points (which consists of all the points for odd values of n) converges to -1, so the only limit points of B are +1 and -1.

(b) Is B a closed set?

B is not a closed set, because NEITHER of the LIMIT POINTS of B is an ELEMENT of B.

To see this, first set [(-1)^n](n)/(n + 1) to +1, and solve for n. That gives [(-1)^n](n) = n + 1.

For all even (and ...

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Georgia Martin, PhD
Education
  • AB, Hood College
  • PhD, The Catholic University of America
  • PhD, The University of Maryland at College Park
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