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We write R+ for the set of positive real numbers. For any positive real number e,...
We write R+ for the set of positive real numbers. For any positive real number e, we write (-6, 6) = {x a real number : -e < x <e}. Prove that the intersection of all such intervals is the set containing zero, n (-e, e) = {0} EER+
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.
5. Describe the following sets of real numbers and find the supremum and infimum of these sets: (a) {x}\x2 – 2 <4€R} (b) {x|x+ 2 +13 – x4<4} (©) {x|x<for all neN} 6. For any two elements x and y of an ordered field, prove that _x+ y + x- x + y - x - y (a) max{x,y}=- (b) min{x,y}=-
[9] Given any two real numbers x and y such that x < y, show that there exists a rational number q such that x < a <y.
Recall that Etan E R is positive if the following two conditions hold: There exists N E Z+ such that an >0 for alln2 N. We use the notation R+to denote the set of positive real numbers: R+ = { E{a») R : Efe») is positive} 1. In class, we proved that the relation<on R, given by is an order relation. In this problem, you'll prove that R satisfies the axioms of an ordered field (a) If E(anh E{놔,Ep., }...
We say that a real number ? is an isolated point of a set ? if ?
is an element of ? and there exists ? > 0
such that ? is the only element of ? that is in the interval (?
− ?, ? + ?)
(a) Prove that every element of the set ? = {1,2,3,… } is an
isolated point of ?.
(b) Prove that if ? is an isolated point of dom(?), then ? is...
Find the cardinality of the set {(r, y) E R? : x² + y? < 1}.
2. Let a be a positive real number, let r be a real number satisfying r >1, let N be an integer greater than one, and let tR -R be the integrable simple function defined such that tr,N(r) = 0 whenver x < a or z > ar*, tr,N(a) = a-2 and tr,N(z) = (ar)-2 whenever arj-ıく < ar] for some integer j satisfying 1 < j < N. Determine the value of JR trN(x) dz.
Real analysis. Please solve all questions
thank you
1. Let h be a positive real number, a <c< d < b and let Sh c< x <d, J() = 1 0 r < c, x > d (a) Using the definition only, find ſº f(x)dx. In fact, given e > 0, you should find an explicit d > 0) which works in the definition. (b) For a given partition P of [a, b], find a good upper bound on S(P)...
O FUNCTIONS AND GRAPHS Union and intersection of intervals B and C are sets of real numbers defined as follows. B={v | v<3) C={v | v>6) Write B U C and B n C using interval notation. If the set is empty, write Ø. BUC- (0,0) [0,0] (0,0) (0,0) DUD BNC = 00 -00 x 5 ? 4 Explanation Check Eng RO tv