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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.
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...
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
5. If a, b E R, prove that abl < (a + b^).
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}=-
Suppose that A is diagonalizable and all eigenvalues of A are
positive real numbers. Prove that det (A) > 0.
(1 point) Suppose that A is diagonalizable and all eigenvalues of A are positive real numbers. Prove that det(A) > 0. Proof: , where the diagonal entries of the diagonal matrix D are Because A is diagonalizable, there is an invertible matrix P such that eigenvalues 11, 12,...,n of A. Since = det(A), and 11 > 0,..., n > 0,...
(10pts) 3. Use direct proof to show that if x and y are positive real numbers, then (x+y)" > " + y".
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.
[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.