Mathematics 713 Assignment #4

    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 sum is less
        than or equal to the sum of the limit suprema.

    2.  Let A and B be bounded subsets of real numbers and define

            A + B = { a+b : a is in A and b is in B }
                                                     _   _   ___
        Prove or find a counter example to the claim A + B = A+B, or
        in otherwords prove or disprove the claim that the closure
        of the sum is equal to the sum of the closures for bounded
        subsets of R.

    3.  Let Fn(x) = x / (1 + n x^2).  Prove or disprove the claim
        that the sequence Fn converges uniformly on R.

    4.  [Extra Credit]  Suppose Pn is a sequence of polynomials for 
        which the degree of Pn is bounded by some finite number M.
        Prove or disprove the claim that if Pn converges uniformly
        to f on [0,1] then f is a polynomial on [0,1].

    5.  McDonald and Weiss problem 3.8.

    6.  McDonald and Weiss problem 3.20 part (a).