3. Prove that (f = 0()] ^ [9 = 0(h)) = f =0(h).
#4 0-0 = -a)|(-0). 3. Prove that for all a, b € 2,00 (-a)0 A 4. Show that for all n e Z, 6]n(n + 1)(2n + 1). 1. 1)) fono
2. Prove that lim (-1)"+1 0. 72-00 n 2n 3. Prove that lim noon + 1 2. 80 4. Prove that lim n-+v5n 0. -7 9 - in 5. Prove that lim n0 8 + 13n 13
1. a) Prove that the axiom "OSz and 0 3 y implies 0 tion is equivalent to "r 3 y and z 2 0 implies xz ya. ry" in Rudin's presenta- b) Prove that if the above axiom is replaced by "O < x and 0 2y implies 0 ry" in the ordered field axiomatic, it follows that 0, b) 1 <0. How do the rules of signs change? c) Would the above replacement make any difference in the axiomatic...
Problem 3. Prove that if bn + B and B < 0, there is an N E N such that for all n > N, bn < B/2.
[3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that \f'(x) < 1/(1 - 1z| for all z e D[0, 1]. [3] 5. Suppose that f: D[0, 1] → D[0, 1] is holomorphic, prove that f'(x) < 1/(1-1-12 for all z e D[0, 1]
m2 2. Prove that lim -+0n3 + 1 -=0. 3 5 100 3n2 + 2n - 1 3. Prove that lim = 5n2 +8 cos(n) 4. Prove that lim = 0. n-700 m2 + 17 5. Prove that lim (Vn+1 - Vn) = 0 Hint: Multiply Vn+1-vñ by 1 in a useful way. In particular, multiply Vn+1-17 by Vn+1+vn
3n+3 3 (i.e. let &>0 and determine a n, to satisfy the definition of convergence.) Prove that lim n5n+5 5 Also, show, using algebraic evidence, that it is an increasing sequence.
Let f(3) = 1 (a) Prove {f} 1 + nx converges to 0 pointwise on (-0,00). (b) Prove or disprove {n} , converges to 0 uniformly on (-0, 0);
(3) 5. Suppose that f : D[0, 1] → D[0, 1] is holomorphic, prove that f'(2) < 1/(1 - 121) for all z e D[0,1].