1. Theorem. (Archimedean Property) Let x be any real number. There exists a positive integer n greater than x.

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1 Advanced Calculus I, Dr. Block, Chapter 2 notes. Theorem. (Archimedean Property) Let x be any real number. There exists a positive integer n greater than x. 2. Definition. A sequence is a real-valued function whose domain consists of all integers which are greater than or equal to some fixed integer (which is often ). The notation { } is used. 3. Definition. We say that a sequence { } converges to a real number L if and only if for every ɛ > 0, there exists a positive integer n such that for all n n we have L < ɛ. The real number L is called the it of the sequence and we write = L. We also say that the sequence is convergent. If there is no real number L as above, we say that the sequence diverges or is divergent. 4. Problem. Prove using the definition that n = 0. Formal Proof. Let ɛ > 0. By the Archimedean Property there exists a positive integer n > ɛ. If n n we have n 0 = n n < ɛ. 5. Problem. Prove using the definition that n 2 + = 0. Preinary consideration: We want n < ɛ for n n. We see that n = Also we will have 5 n < ɛ if n > 5 ɛ. n 2 + n 2 = 5 n. Formal Proof. Let ɛ > 0. By the Archimedean Property there exists a positive integer n > 5 ɛ. If n n we have n = n 2 + n 2 = 5 n 5 n < ɛ. 6. Note. A sequence { } diverges if and only if for every real number L there exists ɛ > 0 such that for every positive integer n there exists n n with L ɛ. 7. Theorem. Any two its of a convergent sequence are the same. (If a sequence converges, then the it of the sequence is unique.) 8. Definition. We say that a sequence { } is bounded if and only if there is a real number B such that B for all n.

2 9. Theorem. Any convergent sequence is bounded. 0. Theorem. If = A and b n = B with A, B R, then. + b n = A + B. 2. b n = A B. 3. b n = A B. a 4. n bn = A B, if B ( ) p = A p, for any positive rational number p, provided that the roots are defined.. Theorem. (Squeeze Theorem) Suppose that { }, {b n }, and {c n } are sequences, and suppose that there exists a positive integer K such that if n K, then b n c n. Suppose that for some real number L Then b n = L. = L = c n. 2. Theorem. If a sequence { } converges to 0 and a sequence {b n } is bounded, then the sequence { b n } converges to Theorem. (Special its to remember and use.). If p > 0, then n p = If r <, then r n = If c > 0, then n c =. 4. n n =. 5. If = 0, then sin( ) = If = 0, then sin( ) =. 4. Definition. We say the sequence { } diverges to if and only if for every M > 0, there is a positive integer n such that for all n n we have > M. In this case we write =. 5. Definition. We say the sequence { } diverges to if and only if for every M < 0, there is a positive integer n such that for all n n we have < M. In this case we write =.

3 6. Theorem. If = and there exists a positive integer K such that b n for all n k, then b n =. 7. Theorem. If = and there exists a positive integer K such that b n for all n k, then b n =. 8. Theorem. Suppose that =.. If {b n } is bounded below, then ( + b n ) =. 2. If {b n } converges or diverges to, then ( + b n ) =. 3. If {b n } is bounded below by a positive number, then ( b n ) =. 4. If {b n } converges to a positive number or diverges to, then ( b n ) =. 5. If {b n } converges to egative number or diverges to, then ( b n ) =. 9. Theorem.. If =, then = If 3. If = 0 and > 0 for all n sufficiently large, then =. = 0 and < 0 for all n sufficiently large, then =. 20. Theorem. (Ratio Test) Suppose that { } is a sequence of nonzero real numbers such that = α + where either α R or α =.. If α <, then = If α >, then =, so the sequence { } diverges. 2. Definition. We say that a sequence { } oscillates if and only if none of the three statements below hold.. = L for some L R.

4 2. =. 3. =. 22. Definition. We say that a sequence { } is increasing if and only if n < k implies a k. 23. Remark. A sequence { } is increasing if and only if for all n we have Remark. A sequence { } of positive real numbers is increasing if and only if for all n we have Definition. We say that a sequence { } is eventually increasing if and only if there is a positive integer n such that n n < k implies a k. 26. Definition. We say that a sequence { } is decreasing if and only if n < k implies a k. 27. Remark. A sequence { } is decreasing if and only if for all n we have Remark. A sequence { } of positive real numbers is decreasing if and only if for all n we have Definition. We say that a sequence { } is eventually decreasing if and only if there is a positive integer n such that n n < k implies a k. 30. Theorem. A bounded, increasing sequence converges. An unbounded, increasing sequence diverges to. 3. Theorem. A bounded, decreasing sequence converges. An unbounded, decreasing sequence diverges to. 32. Definition. We say that a sequence { } is monotone if and only if either { } is increasing or { } is decreasing. 33. Definition. Let ɛ > 0, and let s R. The ɛ-neighborhood of s is N ɛ (s) = {x R : x s < ɛ} = (s ɛ, s + ɛ). The deleted ɛ-neighborhood of s is Nɛ (s) = {x R : 0 < x s < ɛ} = (s ɛ, s) (s, s + ɛ). 34. Definition. Let S R, and let w R. We say that w is an accumulation point of S if and only if every deleted neighborhood of w contains at least one point of S.

5 35. Theorem. Let S R, and let w R. Then w is an accumulation point of S if and only if every neighborhood of w contains infinitely many points of S. 36. Theorem. (Bolzano-Weierstrass Theorem for sets) Every bounded infinite subset of R has at least one accumulation point. 37. Definition. We say that a sequence { } is a Cauchy sequence if and only if for every ɛ > 0, there exists a positive integer n such that for all k, j n we have a k a j < ɛ. 38. Theorem. Let { } be a sequence of real numbers. Then { } is a Cauchy sequence if and only if { } converges. 39. Definition. The sequence {b n } n=i is a subsequence of the sequence {} n=j if and only if there exists a strictly increasing function f : {x N : x i} {x N : x j} such that b n = a f(n) for all n N with n i. We sometimes use the notation b k = k for a subsequence. In this case, n k must be a strictly increasing function of k. 40. Theorem. (Bolzano-Weierstrass Theorem for sequences) Every bounded sequence in R has at least one convergent subsequence. 4. Definition. We let E denote the set of extended real numbers defined by E = R { } { }. 42. Definition. Let { } be a sequence of real numbers, and let A E. We say that A is a subsequential it point of the sequence { } if and only if there is a subsequence k of { } such that k k = A. 43. Theorem. Let { } be a sequence of real numbers. There exists a largest subsequential it point of the sequence and a smallest subsequential it point of the sequence. 44. Definition. Let { } be a sequence of real numbers. The largest subsequential it point of the sequence is denoted by sup. The smallest subsequential it point of the sequence is denoted by inf. 45. Theorem. Let { } be a sequence of real numbers, and let A E. Then = A if and only if A = sup = inf.