1. Let x, a € R. Prove that if a <a, then -a < x <a.
[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]
Let B C R" be any set. Define C = {x € R" | d(x,y) < 1 for some y E B) Show that C is open.
Let Rj be the set of all the positive real numbers less than 1, i.e., R1 = {x|0 < x < 1}. Prove that R1 is uncountable.
Let A, B, C be subsets of U. Prove that If C – B=0 then AN (BUC) < ((A-C)) UB
Question 3 1 pts Let 7 = (xy, - xy) and let D be given by 0 < x <1, 0<y<1. Compute Sap Ē. dr. 0-1 OO O1 O2
help with thus problem but not using schwoz-pick lemma [3] 5. Suppose that f: D[0,1] → D[0,1] is holomorphic, prove that f'(x) < 1/(1 - 1-1) for all z € D[0, 1]
(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].
1 xe Let f(x)={? x 8. Prove that f(x) continuous only at +1. Let f(x)= $3.x xs! x >1 Using the definition prove lim f(x)=1 and lim f (x) = 3 x>17 11°
3. Let D(x) = 18 – 0.4x, S(x) = 3 +0.1x for 0 < x < 40. a. Find the equilibrium point. b. Find the consumer's surplus. C. Find the producer's surplus.