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19 Mar 2008: ii) If fn does not converge uniformly, show that we can find a convergent sequence xn x in[a, b] such that fn(xn) does not converge to f (x). ... derivative’ limh0+ ((f (h) f (0))/h),and similarly for f ′(1)), but such that f ′ is unbounded on [0, 1

10 Mar 2008: Prove that the set S of points x [0, 1] such that x Aj for infinitely many j is dense.Must S be open? ... b) If f : (0, 1) R is continuous and bounded, prove that there exists a sequence of polynomialsQn such that Qn are uniformly bounded and Qn f

19 Mar 2008: n=1 = (x1, x2, x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E, F ) be the space of bounded linear operators equipped with the usual

28 Feb 2008: n}, let the non-negative numbers A(n)k and the“nodes” x(n)k [a, b] be given such that for each polynomial P , the error. ... such that for every f C([0, 1]), there exists a unique series.

14 Oct 2008: If so, howmany such ‘prime triplets’ are there?(b) Suppose that four out of five consecutive odd integers, greater than 10, are prime. ... But such a numberwould clearly be very interesting indeed. Therefore, all positive integers are interesting.

13 Nov 2008: a linear func-tional θ V such that θ(x) 6= θ(y).(b) Suppose that V is finite dimensional. ... If no such subspace exists,show that there exists a basis v, α(v),.

4 Feb 2008: Let a, b K such that thepolynomials f = xm a, g = xm b are irreducible. ... Show that G is solvable if and only if there is an i such that.

28 Aug 2008: anda distinguishedobject e ofL, such that thefollowing hold:. • (a o b) o c = a o (b o c) (associativity). • ... axbaandaxbb. • If m is such that vi a andm b thenm a x b.

28 Aug 2008: anda distinguishedobject e ofL, such that thefollowing hold:. • (a o b) o c = a o (b o c) (associativity). • ... axbaandaxbb. • If m is such that vi a andm b thenm a x b.

23 Feb 2008: 4) Let X be a space, x X, f : S1 X a map such that x is inthe image of f, and Y = X f B. ... 1) Showthat there is an element z C such that γ(z) and x′ in C′.