Just solved: Assign a grade of A (correct), C (partially correct), or F (failure) to
Chapter 7, Problem 21(choose chapter or problem)
Assign a grade of A (correct), C (partially correct), or F (failure) to each.Justify assignments of grades other than A. (a) Claim. Let If and then there is suchthatProof. Let Then so By construction of y, Claim. Let If A is bounded above, then is bounded below.Proof. If A is bounded above, then exists (because iscomplete). Since (see the figure), exists.Thus is bounded below. (c) Claim. If and and both exist, thenProof. Assume and We chooseThen andBy part (ii) of Theorem 7.1.1, there is such thatThen and This is impossible. Therefore,(d) Claim. If f: and A is a bounded subset of then isbounded.Proof. Let m be an upper bound for A. Then for allTherefore, for all Thus is an upper bound for(e) Claim. Let be an ordered field and If thenProof. Suppose Then by property (4) of ordered fieldsso Thus
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