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13 Dec 2001: Then there is some point c in (a,b) such that f'(c) equals (f(b)-f(a))/(b-a). ... x. 2. <. x. n. =b such that the intervals corresponding to neighbouring x.

12 Oct 2023: Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR. 1. Let (zn) be a sequence in R2 such that zn a and zn b. ... On the other hand, show that if fn does not converge uniformly to f thenwe can find a convergent sequence xm x in [a, b] such that fn(xn) 6

6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... If, on the other hand, we cannot find such a b, then A=B and the theorem is proved.

20 Sep 2023: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... Show also that there is a partitionof the vertex-set as X Y such that each of e(X) and e(Y ) is at most 13e(G).

7 Nov 2001: Because of irritating difficulties such as the need to carry digits and to identify 0.999999. ... all y(n) such that y is in B.

15 Jan 2024: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

8 Nov 2023: 12. (a) Show that if x ̸= y are vectors in the finite dimensional vector space V , then there is a linear func-tional θ V such that θ(x) ̸= ... Let A, B V. Prove that A B if and only if Ao Bo.Show that A = V if and only if Ao = {0}.

29 Oct 2018: 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

29 Jan 2002: Once again let us consider a particular example, such as the equation x. ... What we then discovered was that if we take a new number b, given as a root of a polynomial with its coefficients in Q(a) (in such a way that

10 Nov 2006: Does it follow that every open set must be. closed? Give an example of such a metric space. ... Show that thereexists a norm on C(I) such that the map sending f to y0.