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29 Jan 2021: 8. Let t1, t2,. be propositions such that, for every valuation v, there exists n withv(tn) = 1. ... Show that there is a proposition b, inwhich the only primitive propositions appearing are those that appear in both a and c,such that a b and b c.

11 Feb 2014: It follows that for such q, gq λI is positive definite. ... The set of all such t is closed andcontained in [a,b], so it has a minimum value t0.

13 Oct 2006: b) Suppose that f is bounded. Prove that there exists M and n0 such that |fn(x)| < M for all. ... a,b] such that fn(xn) does not converge to f(x). 7. Which of the following functions f are (a) uniformly continuous, (b) bounded on [0,)?

22 Oct 2020: Let B(G,F) be the (extended) Bruhat–Tits building of G(F). For any x B(G,F), there is asmooth affine group scheme Gx over OF such that ... a choice of basis b for Λ such that the embeddings of buildings (3.3.1) and (3.3.2) induceembeddings of the

5 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... Can such a map everbe a Möbius transformation? 9. Let f : U C be a holomorphic function where U is an open set (you may assume f isalso C1).

13 Nov 2002: else prove that no such relation exists. 1. 11. Define a binary operation on Z2 by the formula (a,b) (c,d) = (ac,ad bc). ... a1,b1) (a2,b2). (ak,bk). 12. Let A1,A2,. be sets such that for every positive integer n we have A1.

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.

15 Oct 2021: Define τ by:. U τ for all x U, there exists B β such that x B and B U. ... b) Fix a local base β̇. Show that a sequence (xn)n=1 is τ-Cauchy if and only if for anyB β̇ we can find an integer N such that.

19 Jan 2021: Show that E has a subgroup isomorphic toE/M, and show that all such subgroups are conjugate. ... such that iApA iBpB = idM , pAiA = idA, pBiB = idB, pAiB = 0 andpBiA = 0.

6 Aug 2021: b) Find a sequence fi R([a,b]) such that (fi)iN is Cauchy with respect tothe L2 norm, but for which there is no f R([a,b]) such that ... Proof. Since is open, there exists ,x such that the ball B(x) = {y Rn : |y x| < }is contained in.