State whether each set is bounded below, bounded above, bounded or unbounded for: 1 A =...
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State whether each set is bounded below, bounded above, bounded or unbounded for: 1 A = {r €R|r =, for some n e Z1pts] !! B = {z €R|2 = 2", n € Z}[lpts] Ex.2 Show that every nonempty subset of Z that is bounded above has a max element. [2 pts) Ex.3 If z ¢ Z, show there is exactly one n € Z so that n - 1<z < n. Ex.4 If z - y > 1, z, y €R, show there is at least one n € Z so that y <n < z. Ex.5 If y < z, 1, y €R, show that there is a rational number z such that y <z< r. [2 pts) Ex.6 Find all x € Z so that the following number is an integer: 2 pts) 2 pts (x? – 4)(1- V3) € Z. 2(r + 2) (2 pts) Ex.7 Identify the following sets: A = 0 (z, -), B = U (z,1/2). rE(0,1) zE(0,1) What are the inf, sup, min, max (if they exist) in the case of the two sets above? [2 pts State whether each set is bounded below, bounded above, bounded or unbounded for: 1 A = {r €R|r =, for some n e Z1pts] !! B = {z €R|2 = 2", n € Z}[lpts] Ex.2 Show that every nonempty subset of Z that is bounded above has a max element. [2 pts) Ex.3 If z ¢ Z, show there is exactly one n € Z so that n - 1<z < n. Ex.4 If z - y > 1, z, y €R, show there is at least one n € Z so that y <n < z. Ex.5 If y < z, 1, y €R, show that there is a rational number z such that y <z< r. [2 pts) Ex.6 Find all x € Z so that the following number is an integer: 2 pts) 2 pts (x? – 4)(1- V3) € Z. 2(r + 2) (2 pts) Ex.7 Identify the following sets: A = 0 (z, -), B = U (z,1/2). rE(0,1) zE(0,1) What are the inf, sup, min, max (if they exist) in the case of the two sets above? [2 pts
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