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8 Jan 2021: After allT might prove φ(a) for every a in Γ and yet prove φ(b) for some constant bnot in Γ. ... We need some definitions. DEFINITION 4. A theory T is Γn-complete iff whenever φ(x1,.

15 Oct 2021: for some b > 0, and invoke. the contraction mapping principle]. ...  CbT(1 T) ||w v||X. for some constant C depending on b but not T.

6 Feb 2021: Prove that f is continuous on (a,b). Must it be continuous at a andb too? ... 13. Let f : R R be a function which has the intermediate value property: Iff(a) < c < f(b), then f(x) = c for some x between a and b.

15 Jan 2021: some a,b 0. Hence prove that the class group of K is cyclic and find itsorder.

20 Jan 2021: 6. Let (An : n N) be a sequence of events in some probability space (, F, P). ... P(Xn = 0) 12π. h. (b) Show further that, for all x R,.

15 Oct 2021: Show that ifeb|k|. f̂(k) 2 L2(R) for some b > 0 then f is real analytic at each point in R. ... B to mean that there exists a universal constant C such that A  CB.

7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.

15 Oct 2021: Show that f and g have the same splitting field if andonly if b = cmar for some c K and r N with gcd(r,m) = 1. ... Deduce that every finite abelian group is the Galois group of some Galois extensionK/Q.

15 Oct 2021: for some a < b. ... Lu = u in I, u(a) = u(b) = 0, (?). for some 2 R. Show that the Wronskian:. W(x) = p(x)u0(x)ũ(x) u(x)ũ0(x).

7 Oct 2021: ii) M R n N x S |fn(x)| 6 M. (b) In this case assume that S is a metric space. ... Prove that there is some subinterval [a,b] of [0, 1] with a < b on which f isbounded.