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ANALYSIS II (Michaelmas 2009): EXAMPLES 1 The questions are ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2009-2010/anII-09-1.pdf14 Oct 2009: 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 -
Part 1B Metric and Topological Spaces 2018 pmhw@dpmms.cam.ac.uk…
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2017-2018/MTS1.18.pdf22 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. -
1 Metric & Topological spaces, Sheet 1: 2007 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2006-2007/2007sheet1.pdf8 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′. -
Part 1B Metric and Topological Spaces 2017 pmhw@dpmms.cam.ac.uk…
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2016-2017/MTS1.17.pdf24 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. -
Metric and Topological Spaces Easter 2015 Example Sheet 2 ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2014-2015/MetTop_2015_Ex2.pdf31 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. -
1 Metric & Topological Spaces, sheet 2: 2007 1. ...
https://www.dpmms.cam.ac.uk/study/IB/MetricTopologicalSpaces/2006-2007/2007sheet2.pdf14 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 -
ANALYSIS II EXAMPLES 4 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2005-2006/an05-4.pdf22 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 -
The fundamental theorem of arithmetic
https://www.dpmms.cam.ac.uk/~wtg10/FTA.html15 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. -
ANALYSIS II EXAMPLES 1 Michaelmas 2004 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2004-2005/an04-1.pdf21 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 -
ANALYSIS II EXAMPLES 1 Michaelmas 2005 J. M. E. ...
https://www.dpmms.cam.ac.uk/study/IB/AnalysisII/2005-2006/an05-1.pdf18 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
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