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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

11 Oct 2023: Let A and B be subspaces of a vector space V such that V = A B, and let ‖.‖1and. ... Construct two normed spaces X and Y such that d(X, Y ) = 1 but X and Y are not.

21 May 2023: Using (a) and (b), or otherwise, show that for all pairs of positive naturalnumbers ⟨m, n⟩, there is a natural number h such that ⟨m, n⟩ ⟨h, h⟩.

16 Nov 2023: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

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

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).

1 Mar 2023: i) f(z)/z 0 as |z|.(ii) There exists b C and > 0 such that for every z C, |f(z) b| >. (iii) f = u iv and |u(z)| > ... 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.

10 Oct 2023: Suppose that α has minimal polynomial f over Q, and let β = α4.Find a, b, c Q such that β = a bα cα2. ... 9. Let K and L be subfields of a field M such that M/K is finite.

29 May 2023: you definedisjoint union operationally by describing what it does, by giving a spec; youimplement it in such a way that it fits the spec). ... And we have to cook up such a function using only f, a and b.

21 Nov 2023: Let a,b K such that the polynomials f = Xm a, g = Xm b are irreducible. ... Use asuitable Lagrange resolvent to find a Q(ζ3) such that K = Q(ζ3, 3.