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25 Jun 2015: Foreach open set A Ψ(U )cs and each M A, we shall show that there is a smalleropen neighbourhood M B A Ψ(U )cs such that. ... Then we may choose smooth functions ϕi : M [0, 1] such that supp(ϕi) int(Vi) and p(ν K) (. ϕi)1(1) and choose B =. B(A, ϕ1

9 Feb 2015: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

20 Nov 2015: If A is the matrix of the first of these (say), find a non-singular matrix P such that PTAP is diagonalwith entries 1. ... 14. Let a1, a2,. , an be real numbers such that a1 an = 0 and a21 a2n = 1.

28 Aug 2015: p. k, where k is the smallest value ofn such that xn 6= yn. ... Hint:first find a subsequence (fni ) such that fni (x) converges for all x Q [0, 1].)Deduce that the closure of B in (C[0, 1], d) is compact.

26 Nov 2015: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... 6. Show that every graph G has a partition of its vertex-set as X Y such that the numberof edges from X to Y is at least 12e(G).

27 Jan 2015: a). n=1. nenz on {z. 0 < r Re(z)}; (b) n=1. ... Can such a map ever be a Möbius trans-formation? 9 Calculateγ z sin z dz when γ is the straight line joining 0 to i.

26 Oct 2015: fn for V such that. ψ.  mi=1. xiei,. nj=1. yjfj. ... Suppose thatα: V W is a linear map such that α(Y ) Z.

11 Feb 2015: If X is a space such that H(X; R) = H(Sn; R), show thatthere is a map f : Sn X inducing an isomorphism on H(; R), and hence thatπi(X) ... Hint: a) Show that such rings have Tor(,R) = 0; b)Consider the class of abelian groups A such that AR = 0.].

9 Feb 2015: That is, for everya < b and every t there should exist x (a,b) such that f(x) = t. ... 14. Let f : R R be a function that has the intermediate value property: that is, iff(a) < c < f(b) then there exists x (a,b) such that f(x) =

21 Oct 2015: a) Let a G. Show that there is a positive integer n such that an = e, the identity element.(The least such positive n is the order of a.). ... b) Show that there exists a positive integer n such that an = e for all a G.(The least such positive n is the

2 Nov 2015: Fix j, let > 0 and choose a partition P = {a = a0 < a1 <. < an = b} such thatU(P,f) L(P,f) < j1. Let K = {k : Ej (ak,ak1) 6= }. Then Ej ... an = b} be any partition of [a,b] such thataj1 aj < δ, and let J = {j : [aj,aj1] F 6= }. Show that sup[aj,aj1] f

19 Feb 2015: Deducethe more general result that if f ′(a) 6= f ′(b) and z lies between f ′(a) and f ′(b), then thereexists c (a,b) such that f ′(c) = z. ... f(a) < 0 and f(b) > 0 then there exists c (a,b)such that f(c) = 0.

16 Oct 2015: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a,b] such that fn(xn) 6 ... a) Must there exist a sequence (fn) of continuous functions on [0, 1] such that fn f uniformly on [0, 1]?(b)?

10 Feb 2015: Show that for any b C, there exists a sequence of points zn D(a,R) with zn 6= a such thatzn a and f(zn) b as n. ... Find such a sequence when f(z) = e1/z, a = 0 and b = 2.

20 Nov 2015: Can S be uncountable?Is there an uncountable family T PN such that AB is finite for all distinct A, B T? ... You are asked to supply them with a strategy such that,should they follow it, only finitely many of them will guess wrongly.

15 Oct 2015: 1. Calculate d = (a,b) and find integers r and s such that ra sb = d when. ... iii) Find constants c and d such that λ(a,b) 6 c log b d.

4 Mar 2015: Prove that f is. integrable. What is 10f(x)dx? 6. Let a < b and let f : [a,b] R be a Riemann integrable function such that f(x) 0for every ... 1, 1], such that f(x) = g(x) h(x) for every x.

13 Nov 2015: every polynomial p of degree n there is x0 [0, 1/n]such that |p(x)| C|p(x0)| for every x [0, 1].(b) If (X,d) is a metric ... That is to say, showthat there is no metric d on the set of continuous functions f : [a,b] R such that pointwise convergenceof

14 Oct 2015: algebra. 1.2. Show that the following sets of subsets of R all generate the same σ-algebra:(a) {(a,b) : a < b}, (b) {(a,b] : a < b}, (c) {(,b] : b ... an,bn] such that the Lebesgue measure of A4B(= (Ac B) (ABc)) is less than ε.

30 Nov 2015: Does it follow that thereexists c (a,b) such that f(b) f(a) = f′(c)(ba)? ... containing (x0,y0) and V containing F(x0,y0) = (x0, 0) such that F maps U bijectively to.