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10 Mar 2017: Lent 2017 ANALYSIS I – EXAMPLES 4 AZ. 1. Let f : [a,b] R be an integrable function with f(x) > 0 for every x.Assume further that f is continuous. ... Show that for every. closed subinterval I [a,b] of positive length and every ε > 0 there existsa

31 Oct 2017: Dissertation. 22. Except as provided in (a) and (b) below, every candidate for Part II shall send to the Chair of Examiners so as to arrive not later than the first ... 23. Every candidate submitting a dissertation or a critical review essay under

27 Jun 2017: We say that B £ co is a cover of J / , if /I £ B for every X e J /. The family <E/ is called p-closed, if for every countable ... b) Every ideal of character < d can be extended to a nonmeagre p-ideal of.

10 Oct 2017: Show that G is infinite, but that every element a of G has finite order. ... 16. Show that every isometry of C is either of the form z 7 az b or the form z 7 az̄ b with a,b C and|a| = 1 in

5 Feb 2017: Show that there exist unique complexnumbers A, B such that for every z C, T(z) = Az Bz̄. ... closed andbounded) subsets of the given domains in C:. (a)n=1. nenz on {z : 0 < Re(z)}; (b).

13 Dec 2017: Show that. (a) f, D1f, and D2f are continuous in R2.(b) D12f and D21f exist at every point in R2 and are continuous except at (0, 0).(c) ... 13. Let U R2 be an open set containing the rectangle [a, b][c, d].

27 Oct 2017: 1. Every Supplicat, that is to say, every Grace for a complete degree not conferred under Statute B II 2, shall be in the form prescribed from time to time by ... a)every titular degree shall be conferred before any complete degree is conferred;.

10 Feb 2017: Prove that every r nLatin rectangle may be extended to an n n Latin square.7. ... Show that a graph G (with |G| > k) is k-connected if and onlyif there is an a-B fan for every B V (G) with |B| = k and every vertex

5 Feb 2017: every z C, |f(z) b| > ε.(iii) f = u iv and |u(z)| > |v(z)| for all z C.4. ... Showthat f(z) = z.(ii) Let f be an entire function with f(n) = n2 for every n Z.

22 Oct 2017: Hint: Bolzano-Weierstrass.). 3. If A and B are subsets of Rn, let A B = {a b |a A,b B}. ... vn} C such that{B(v1),. ,B(vn)} cover C. (b) Show that there exists some > 0 such that for every v C, B(v) Uα for some α.Deduce that