Mathematics 713 Assignment #3

    1.  Let Xn and Yn be bounded sequences of real numbers.  Prove
        or disprove the claim that

            limsup Xn Yn ≤ (limsup Xn)(limsup Yn)

        in otherwords that the limit supremum of the product is less
        than or equal to the product of the limit suprema.

    2.  Let E be a subset of R and E' be the set accumulation points
        as defined in exercise 2.43 of McDonald and Weiss.  Prove or
        disprove the claim that
            _
            E \ E'

        is closed, or in otherwords that the set of real limit points
        of E which are not accumulation points is closed.
                                                   _   _
    3.  Let A and B be open subsets of R such that A = B, that is,
        such that their closures are equal.  Prove or disprove the
        claim that A = B.

    4.  [Extra Credit]  Let f be a real valued function defined on
        the interval [a,b].  Let 

            B = { c : lim  f(x) = L exists but L ≠ f(c) }
                      x→c    

        be the set of removable discontinuities of f.  Prove or
        disprove the claim that B is a countable set.

    5.  McDonald and Weiss problem 2.54.

    6.  McDonald and Weiss problem 2.85.