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10 May 2002: A formal definition could be something like this. A multivalued function from A to B is a subset H of AxB such that for each a in A there is at ... least one b in B such that (a,b) is in H.

19 Feb 2024: 8. Find two totally ordered sets such that neither is isomorphic to a subset of the other.Can you find three such sets? ... Is there a countable ordinal α such thatα is the αth fixed point?

22 Nov 2012: Show that, for such a map,. dimH f (M ) 61α. ... Prove that the limit set for such a group is totallydisconnected.

24 Nov 2001: 8. A real number is a partition of the rational numbers into two sets A and B such that every element of A is less than every element of B. ... B such that (a,b) is in F.

11 Oct 2023: Let A and B be subspaces of a vector space V such that V = A B, and let ‖.‖1and. ... Construct two normed spaces X and Y such that d(X, Y ) = 1 but X and Y are not.

8 May 2003: n. ) with certain properties, such that a. n. tends to infinity, then one can first try to get every a. ... n. ) such that b. n. /a. n. tends to zero and Sum.

8 Feb 2024: i) f(z)/z 0 as |z|.(ii) There exists b C and > 0 such that for every z C, |f(z) b| >. (iii) f = u iv and |u(z)| > ... 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.

4 Oct 2001: Of course, we could say something like there exists a constant lambda such that a. ... x[y]-y[x]]. 2. =0. I put the square in because one always likes to expand such an expression in terms of inner products.

16 Nov 2023: 13. Let S be a collection of subsets of N such that for every A,B S we have A B or B A.Can S be uncountable? ... Is there an uncountable collection T of subsets of N such that A B isfinite for all distinct A,B T?

26 Mar 2003: Let A be a set that is NOT. Then we can find a subset B of {1,2,.,N} with |B|> cN and a function g:B-->C such that ... Zeilberger should not be disappointed if attempts to find such a proof eventually succeed.

8 Feb 2024: Regarding T as a map from C into C, show that there existunique complex numbers A,B such that for every z C, T(z) = Az Bz. ... Can such a map be the restriction tothe strip of a Möbius map?

13 Dec 2001: Then there is some point c in (a,b) such that f'(c) equals (f(b)-f(a))/(b-a). ... x. 2. <. x. n. =b such that the intervals corresponding to neighbouring x.

12 Oct 2023: Mich. 2023 ANALYSIS AND TOPOLOGY—EXAMPLES 1 PAR. 1. Let (zn) be a sequence in R2 such that zn a and zn b. ... On the other hand, show that if fn does not converge uniformly to f thenwe can find a convergent sequence xm x in [a, b] such that fn(xn) 6

7 Nov 2001: Because of irritating difficulties such as the need to carry digits and to identify 0.999999. ... all y(n) such that y is in B.

20 Sep 2023: Provethat A contains an infinite set B such that no point of B is a convex combination of otherpoints of B. ... Show also that there is a partitionof the vertex-set as X Y such that each of e(X) and e(Y ) is at most 13e(G).

6 Mar 2002: b in B (that is, some non-zero b (mod p)) such that the set of all bx. ... If, on the other hand, we cannot find such a b, then A=B and the theorem is proved.

15 Jan 2024: 6. Let R be an integral domain. The greatest common divisor (gcd) of non-zero elements a andb in R is an element d in R such that d divides both ... b) Show that may be identified with the set Fq{} in such a way that GL2(Fq) acts on as the group of

8 Nov 2023: 12. (a) Show that if x ̸= y are vectors in the finite dimensional vector space V , then there is a linear func-tional θ V such that θ(x) ̸= ... Let A, B V. Prove that A B if and only if Ao Bo.Show that A = V if and only if Ao = {0}.

29 Jan 2002: Once again let us consider a particular example, such as the equation x. ... What we then discovered was that if we take a new number b, given as a root of a polynomial with its coefficients in Q(a) (in such a way that

12 Oct 2007: On the other hand, show that if fn does notconverge uniformly to f then we can find a convergent sequence xm x in [a, b] such that fn(xn) 6 f ... b) Give an example of a sequence of uniformly continuous, real-valued functions (fn) on R such thatfn