Exercise 1.25. This exercise relates to (1.13). Suppose that x > 1. For each ne N...
Finish the proof of Theorem 3.14.
Theorem 3.14 Let (neN aand EneN be sequences in R. Let be in R# and suppose that x" → x, y, → oo, and z" →-oo. . If -oo <x o, then +yn 2. If-oo x < 00, then x" + Zn →-00 4. If-oo x < 0, then xoY" →-00 and xnZn → oo. 5. If x is in R. then-→0and-" →0 Proof Note that the conditions in the different parts of the...
[ARCHIMEDES] Suppose that Xo = 2/3, yo = 3, xn = 2xn-1 Yn-1 xn-1 + Yn-1 and Yn = 1xn Yn-1 for ne N. a) Prove that xnx and Yn 1 y, as n = , for some x, y E R. b) Prove that x = y and 3.14155 < x < 3.14161. (The actual value of x is a.)
2.21 Let Q(2) = VI, which is defined for all x > 0. Prove: Q E C[0,0). (Hint: If a > 0, and € > 0, we seek 6 >0 such that 3 > 0 and - al <& implies Q(x) - Q(a) < €. Begin by showing that|vx-Val' <lt - al.)
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...
that h(mn ) h ( m)n, h ( ) and that if m < n then h ( m ) < n ( n ) = . Exercise 2.7.4. [Used in Theorem 2.7.1.] Complete the missing part of Step 3 of the proof of Theorem 2.7.1. That is, prove that k is surjective. Exercise 2.7.5. [Used in Theorem 2.7.1.] Let Ri and R2 be ordered fields that satisf We were unable to transcribe this imageWe were unable to transcribe this...
ly(mod n). 2. Let n > 1 be an odd integer and suppose ? = y2 (mod n) for some x Prove that ged(x - yn) and ged(x + y, n) are nontrivial divisors of n.
all three questions please. thank you
Prove that for all n N, O <In < 1. Prove by induction that for all n EN, ER EQ. Prove that in} is convergent and find its limit l. The goal of this exercise is to prove that [0, 1] nQ is not closed. Let In} be a recursive sequence defined by In+1 = -) for n > 1, and x = 1. Prove that for all ne N, 0 <In < 1....
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show that P(Yn = 1) = 1/2, but Yn, n E N is not a Bernoulli process
Exercise 2. Let Xn, n EN, be a Bernoulli process uith parameter p = 1/2. Define N = min(n > 1:X,メ } For any n 2 1, define Yn = XN4n-2. Show...
1. Let {rn;n > 1} be a sequence of real numbers such that rn → x, where r is real. For each n let yn = (1/n) E*j. Show that yn + x. HINT: (xj – a) Let e >0 and use the definition of convergence. Split the summation into two parts and show that each is < e for all sufficiently large n.
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x, y, 0) = (0, y, a) orientation-preserving or
EXERCISE 3.55. Let V denote the ry-plane in R3, oriented by N : - (0,0, 1) function f: V V defined as orientation-reversing? Let V denote the yz-plane, oriented by N = (-1,0,0). Is the f(x,...