lf {Ζ., n>0) is a martingale, is this true for E[Zw | Z.]= Zn for n>0? Please prove. d lf {Ζ., n>0) i...
Number Theory 13 and 14 please! 13)) Let n E N, and let ā, x, y E Zn. Prove that if ā + x = ā + y, then x-y. 14. In this exercise, you will prove that the additive inverse of any element of Z, is unique. (In fact, this is true not only in Z, but in any ring, as we prove in the Appendix on the Student Companion Website.) Let n E N, and let aE Z...
8. Suppose b 2 a z 0 for all k E N. a. lf 2. bconverges, prove that £f.i ai converges b. If Ea diverges, prove that £** b, diverges.
Suppose that a sequence {Zn} satisfies Izn+1-Znl < 2-n for all n e N. Prove that {z.) is Cauchy. Is this result true under the condition Irn +1-Fml < rt Let xi = 1 and xn +1 = (Zn + 1)/3 for all n e N. Find the first five terms in this sequence. Use induction to show that rn > 1/2 for all n and find the limit N. Prove that this sequence is non-increasing, convergent,
0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT 0 and 0, and let a E Z. Prove that [a],m C [a]n if and only if n | Let m,EN with m TT
9·Let m, n E Z+ with (m, n) 1. Let f : Zmn-t Zrn x Zn by, for all a є z /([a]mn) = ([a]rn , [a]n). (a) Prove that f is well-defined. (b) Let m- 4 and n - 7. Find a Z such that f ([al28) (34,(517). (c) Prove that f is a bijection.2 (HINT: To prove that f is onto, given (bm, [cm) E Zm x Zn, consider z - cmr + bns, where 1 mr +ns.)
(3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'| <E (3) Suppose that f E C'((0, 1]). Given e > 0, prove that there exists a polynomial p such that lf-plloo -p'|
Please only do number 8. 7. Prove that if {zn} is a sequence with lim zn w and lim zn = w, then lim z 8. For the sequence of the previous exercise, prove that lim 1/zn = 1/w provided w 0.
8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u < o0. Let Z, = ... Хо. X. 8.33 Let X1, X2, L(X — м), for n 3D 0, 1, 2, .. (a) Show that Zo, Z\, ... is a martingale with respect to be i.d random variables with mean u
5. For z, w E C, show the following identities. (a) z + w = z + W (b) zw = zw (c) |zw| = |2||w (d) () = 1 where w #0 (e) [2"| = |z|" where n is a positive or negative integer
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C (b) Prove that when z є R, the definition of exp z given above is consistent with the one given in problem (2a), assignment 16. Definition from Problem (2a): L(x(1/t)dt E(z) = L-1 (z) 2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C...