# Archimedean Principle

Midterm Exam, Math 370 There are four questions on the exam. The first two questions are worth thirty points each. The last two are worth twenty points each. There are 100 points on the exam. 1. Do TWO of the following: a) State the Archimedean Principle. b) State the Bolzano-Weierstrass Theorem. c) What does it mean for a sequence {xn } to be Cauchy? d) State the Squeeze Theorem for sequences. 2. Are the following true or false? Justify your answers briefly. a) If x, y, z are real numbers, and x < y, then xz < yz. b) If {xn } is a sequence with xn < 3 for all n, then it cannot converge to 3. c) Suppose {xn } is a sequence with the following property: ∀ > 0, ∃N ∈ N such that n ≥ N ⇒ xn < . Then xn → 0. : n ∈ N} exists. d) sup{ n−1 n 3. Let {xn } be a sequence with the property that there exists N0 ∈ N and c ∈ R such that n ≥ N0 ⇒ xn = c. Show that {xn } converges and find its limit. 4. Define the sequence {xn } as follows. Let x1 = 1, and for n ≥ 1, 7xn+1 = x2n + 10. a) Show that 2 < xn < 5 for n ≥ 1. (HINT: Use induction) b) Show that {xn } is decreasing. (HINT: Consider xn+1 − xn ) c) Show that {xn } converges and find its limit. Midterm Exam, Math 370 There are four questions on the exam. Each question is worth ten points. 1. Do TWO of the following: a) What does it mean for a sequence to be Cauchy? b) State the Bolzano-Weierstrass Theorem. c) What does it mean for a sequence {xn } to converge to a real number a? d) State the Squeeze Theorem for sequences. 2. Are the following true or false? Justify your answers briefly. a) If {xn } is a bounded sequence, then it is Cauchy. b) If |xn | → |x|, then xn → x. c) Let f be any function defined on R. If limx→0+ f (x) = 1, then limx→0 f (x) = 1. d) Let a ∈ R. If xn → a and yn → a, then xn = yn for all n ∈ N. 3. Let {sn } and {an } be sequences, and let s ∈ R. Suppose there exists K > 0 such that |sn − s| ≤ K|an | for all n ∈ N. Show that if an → 0, then sn → s. 4. Let f (x) = x2 + 2x + 6. Show that limx→3 f (x) = 21. Math 370 Midterm Exam March 18, 2021 There are four questions on the exam. Each question is worth ten points. 1. Do TWO of the following: a) What does it mean for a sequence to be Cauchy? b) State the Bolzano-Weierstrass Theorem. c) What does it mean for a sequence {xn } to converge to a real number a? d) State the Squeeze Theorem for sequences. 2. Are the following true or false? Justify your answers briefly. a) If xn > 0 for all n ∈ N , and xn → x, then x > 0. b) If {xn } is a sequence such that {x2n } converges, then {xn } converges. c) If {xn } is Cauchy, then it is increasing. d) For any positive number r, {rn } converges. 3. Suppose xn → a, and {yn } is a sequence with the property that |xn −yn | ≤ for all n ∈ N . Show that yn → a. 1 1 n 4. Let f (x) = 3 sin x . x2 + 1 Show, using the definition, that limx→0 f (x) = 0. (HINT: x2 + 1 ≥ 1 and | sin x| ≤ |x| for all real numbers x.) 2
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