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22 Mar 2018: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

14 May 2003: Proof. Applying the hypothesis about b to the bilinear map g:UxV-->U@V, we obtain a linear map i:X-->U@V such that g=ib. ... Similarly, applying the universal property of g to the bilinear map b, we obtain a linear map j:U@V-->X such that b=jg.

8 May 2007: b) Give a topology on R which is neither trivial nor discrete such that every open set isclosed (and vice-versa). ... b) (x, y, z) (x′, y′, z′) x = x′, y = y′, z = 1 z′.

24 Apr 2017: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

31 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.

14 May 2007: Is there always an orientable such surface, i.e. one in which“turning clockwise” makes sense globally on the surface? ... b) What is the smallest genus (number of holes) of such a surface with boundary thetrefoil knot, which is the knot which lies on

22 Nov 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 9. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the

15 Jun 2000: We must convert a into a new number a' in such a way that p divides a'b and does not divide a'. ... Then we can find H and T such that HaTb=1, and we know that we will now find that p divides b if we continue the argument.

21 May 2005: a,b] such that fn(xn) does not converge to f(x). [Hint: Bolzano–Weierstrass.]. ... every point of [0,1] (here we interpret f ′(0) as the ‘one-sided derivative’ limh0+((f(h) f(0))/h),and similarly for f′(1)), but such that f ′ is

18 Oct 2005: ii) Give an example of a sequence of uniformly continuous functions fn on R, such that fn con-. ... a,b] such that fn(xn) does not converge to f(x). 12. (i) Suppose f is defined and differentiable on a (bounded or unbounded) interval E R, andthat its

15 Oct 2010: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a, b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

23 Jun 2012: 2) Let (Rn,d) denote Euclidean n-space. If P,Q,R are points in Rn such that. ... For any sequence (gn) Y such that gn g X, show that g Y.

21 May 2005: iii) Give an example of disjoint closed sets A and B in R2 with points an A and bn B such. ... 10. [Tripos IB 96401(b), modified] (i) Suppose that (X,d) is a nonempty complete metric space, andf : X X a continuous map such that, for any x,y X, the

18 Oct 2011: On the other hand, show that if fn does not converge uniformly to f, then we can find aconvergent sequence xm x in [a,b] such that fn(xn) does not ... Give an example of a sequence of uniformly continuousfunctions fn on R, such that fn converges

7 Mar 2024: b) Find a condition on 𝑠 such that 𝛿𝑥 𝐻𝑠(R𝑛). c) Show that 𝐻𝑡 (R𝑛) is continuously embedded in 𝐻𝑠(R𝑛) for 𝑠 < 𝑡. ... Δ2𝑢 𝑢 = 𝑓. b) Show further that there exists 𝐶 > 0 such that ‖𝑢‖𝐻 𝑠4

20 Oct 2014: metric relation R defined on X such that for any k elements x1,. ... Prove that there existu,v Zp such that. |A (uB v)| αβp αβp, where uB v denotes the set{ub v : b B}.

4 May 2006: Prove that f has a real root. 17. Prove there is no continuous function f : [0, 1] R such that x Q f (x) 6 Q (whereQ denotes the rational numbers). ... 18. A continuous surjective map p : E B is a covering if every point b B has an openneighbourhood V

11 Mar 2007: Then there is asubset B A of size β such that all the unordered γ-tuples from it are in thesame piece of the partition. ... an1, ν〉 such that 〈a1. an1, ν, c〉 and 〈a1. an1, ν, c′〉 are coloureddifferently by.

21 May 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the operator

4 Nov 2005: n=1 = (x1,x2,x3, ) such that for each n, |xn| 1/n. ... In such circumstances, T is a bounded linear operator. (What is the moral of this equivalence?)Let B(E,F) be the space of bounded linear operators equipped with the usual