to restate part (ii). Lemma 1.3.8. Assume s ER is an upper bound for a set A CR. Then, sup A if and only if, for every choice of e> 0, there exists an element a Є A satisfying s - € < a. 18 Chapter 1. The Real Numbers Proof. Here is a short rephrasing of the lemma: Given that s is an upper bound, s is the least upper bound if and only if any number smaller than s is not an upper bound. Putting it this way almost qualifies as a proof, but we will expand on what exactly is being said in each direction. () For the forward direction, we assume s = sup A and consider s-e, where € > 0 has been arbitrarily chosen. Because s-e 0 is chosen, s-e is no longer an upper bound for A. Notice that what this implies is that if b is any number less than s, then b is not an upper bound. (Just let e = s-b.) To prove that s = sup A, we must verify part (ii) of Definition 1.3.2. (Read it again.) Because we have just argued that any number smaller than s cannot be an upper bound, it follows that if b is some other upper bound for A, then s sup B. (b) A finite set that contains its infimum but not its supremum. (c) A bounded subset of Q that contains its supremum but not its infimum. Exercise 1.3.3. (a) Let A be nonempty and bounded below, and define B = {bЄR: b is a lower bound for A}. Show that sup B = inf A. (b) Use (a) to explain why there is no need to assert that greatest lower bounds exist as part of the Axiom of Completeness. Exercise 1.3.4. Let A1, A2, A3,... be a collection of nonempty sets, each of which is bounded above.

An apple farm yields an average of 45 bushels of apples per tree when 21 trees are planted on an acre of ground. Each time 1 more tree is planted per acre, the yield decreases by 1 bushel (bu) per tree as a result of crowding. How many trees should be planted on an acre in order to get the highest yield? In order to get the highest yield, trees should be planted on an acre. OCEED

Please answer a-d. Thanks

The first Batman movie was made over 50 years ago in 1966. Över the years, Batman has been played on screen by a number of actors and even by a Lego figure in the Lego Batman movies. In the original comic books, Batman was 188 centimeters tall (about 6'2") and weighed 95 kilograms (about 210 pounds). The article "50 Years of Batman on Film: How Has His Physique Changed?"t included the heights and weights of all of the onscreen Batmen in the table below. Batman Height (cm) Weight (kg) Comic book 188 95 Lego Batman 4 0.005 Adam West 188 91 Michael Keaton 178 72 Val Kilmer 183 93 George Clooney 180 78 Christian Bale 183 82 Ben Affleck 193 98 A USE SALT With x = height and y = weight, the equation of the least squares regression line is ŷ = -3.0112 + 0.4881x. %D (a) Calculate the residuals. (Round your answers to two decimal places.)