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Prove that {19a +37b| : a,b E Z} = NU {0}. Prove that {19a +37b| : a,b E Z} = NU {0}.
Let f Ecº [a, b] such that [ f(x)z”dx = 0 for all n E NU {0} Prove that f(x)dt = 0 conclude that f(x) = 0 for all r = [0, 1]
(2) Prove that if j-0 i-0 with k, 1 e N u {0), and bo, . . . , be , do, . . . , dl e { 0, . . . , 9), such that be, de # 0, then k = 1 and bi- di fori 0,.. , k. (I recommend using strong induction and uniqueness of the expression n=10 . a + r with a e Z and re(0, 1, ,9).) (3) Prove that for all...
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...
definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0
definition of limit to prove that lim ,-e3. 3, (a) Use the - (b) Suppose lim g(z) 0 and if(x)| |g(z)| for all z E R. Use the ε-δ definition of limit to prove that lim f(x)=0
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
Problem 5. (i) Prove that sin (5) if 0 < If z = 0 £1 f(z) = 1。 is Riemann integrable on 0, (ii) Prove that if z if z E {0, π, 2r) g(z) = 0 is Riemann integrable on [0,2
(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0
(14.3) Suppose that f()-OP0cman for z E C. Prove that, for all R. where ) n=0
(3) If z = a + ib E C and |2| := Va² + b², prove that |zw| = |z||w]. Proof. Proof here. goes (4) Let y : C× → R* be defined by 9(z) = |z|. Use Problem (3) to prove that y is a homomorphism. Proof. Proof goes here.
Prove that A = B for: A = {(x,y) e Rº : +y/<1} B = {(z,y) € RP: (71+ y)² < 1}
10. Let a, b,n E Z such that n >0, n does not divide a and al B in Z/nZ. Assume a-and [N]-[a]. Prove n #313 and n 497, 4