P.6. Problem Set (Final Review)

P.6. Problem Set (Final Review) - Math 311 Review Problems...

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Math 311: Review Problems for Final #1 Let S be a nonempty subset of real numbers that is bounded above. Put . Prove that for every , there is some x in S satisfying . #2 Let A and B be nonempty subsets of real numbers that are bounded above. Then it is appropriate to put and . Prove that where . Is it true that ? Justify your answer. #3 Let S be a nonempty subset of real numbers that is bounded above. Put a. Show that where . b. Show that where #4 Give an proof that converges to . #5 State and prove the Squeeze Theorem for sequences #6 State and prove the Nested Interval Theorem #7 State and prove the Monotone Convergence Theorem #8 Let x be any positive real number and define a sequence by where represent the largest integer less than or equal to x , i.e. . Prove that converges to x /2. #9 (Cesaro Means) Show that if is a convergent sequence, then the sequence defined by the arithmetic means must converge to the same limit. Give an example where the sequence diverges but the sequence converges.
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