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8 May 2003: n. ) with certain properties, such that a. n. tends to infinity, then one can first try to get every a. ... n. ) such that b. n. /a. n. tends to zero and Sum.

29 Sep 2003: Tripos IB 96401(b), modified] (i) Let (X,d) be a nonempty complete metric space, and let f :X X be a continuous map such that, for any two points ... xn} is a finite set of points inX such that the balls B(xi,) cover X, consider the 2n 1 points of HX

25 Jul 2003: 1. such that g(n2)(cn21 ) = g(n2)(cn22 ) = = g(n2)(cn2n1) = 0. ... From Lemma 6.5. there. exists a function g C(T) such that ‖g‖ < 1 and |SM(g, 0)| B log M > Kε.

26 Mar 2003: Let A be a set that is NOT. Then we can find a subset B of {1,2,.,N} with |B|> cN and a function g:B-->C such that ... Zeilberger should not be disappointed if attempts to find such a proof eventually succeed.

29 Sep 2003: 1. 2 ANALYSIS II EXAMPLES 1. (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 ... derivative’ limh0+ ((f(h) f(0))/h),and similarly for f′(1)), but such that f′ is unbounded on [0,

21 Aug 2003: in the case that the underlying monad supports such anmfix operation [9, 22]. ... We consider a Conway operator on such a C, that is, a matrix Conway theoryfor Bloom and Ésik [4].

17 Jul 2003: E/K ) contains non-trivial. n-torsion. Such calculations are referred to as descent calculations. ... Furthermore there is a pairing : [A] [B] Z/nZ such that S(φ)(Cλ/Q)=kerL () and S(̂φ)(Dλ/Q)= kerR().

24 Nov 2003: Deducethat there exists a pair of irrational numbers a, b such that ab is rational. ... 3. 2 100. 3. 2 is irrational. 19. Construct a function f : R R that takes every value on every interval - in otherwords, such that for every a < b and every

19 May 2003: matrix P and a non singularn n matrix Q such that B = QAP. ... Two n nmatrices A and B can represent the same endomorphism of V (with respectto appropriately chosen bases) if and only if there exists a non singular nnmatrix P such that

28 Nov 2003: 1. Does there exist an uncountable family B of subsets of N such that for every A,B B(distinct) the intersection of A with B is finite? ... a1,a0,a1,a2,. be a doubly infinite sequence of points in [0, 1] such that for.

9 Jun 2003: To give a simple example: if I take two subsets A and B, I can define their union to be the set of all n such that n belongs either to ... a. 1. ,b. 1. ] is any closed interval such that a.

2 Aug 2003: nk=0(t xk) which is small for all t [a,b]. Such a polynomial was. ... 12. Lemma 5.2. If f : [a,b] C is continuous then we can find an x1 [a,b]such that.

10 Sep 2003: x), and such thatddtρ(t) = Y (ρ(t)) at the points ρ(t) U. ... B. Assume the claim for the moment. It follows that there exists a nonzero numberα such that:.

5 Nov 2003: AxB, the Cartesian product of A and B, is defined to be the set of all ordered pairs (x,y) such that x is an element of A and y is ... possible such that r is non-negative) with the same highest common factor.

3 Sep 2003: n. ) and (b. n. ) tending to root two from below and above respectively in such a way that f(b. ... there must be some c between a and b such that f(c)=w.

14 May 2003: Proof. Applying the hypothesis about b to the bilinear map g:UxV-->U@V, we obtain a linear map i:X-->U@V such that g=ib. ... Similarly, applying the universal property of g to the bilinear map b, we obtain a linear map j:U@V-->X such that b=jg.