Explore BrainMass

Explore BrainMass

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

    This content was COPIED from BrainMass.com - View the original, and get the already-completed solution here!

    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.

    © BrainMass Inc. brainmass.com March 4, 2021, 9:49 pm ad1c9bdddf
    https://brainmass.com/math/real-analysis/limit-points-of-set-closure-of-set-isolated-points-of-set-274347

    Attachments

    Solution Preview

    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 ...

    Solution Summary

    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.

    $2.49

    ADVERTISEMENT