# In this case we say that s n is convergent we say

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In this case we say that s n is convergent. We say that lim n →∞ s n = is for any number A , there is some integer N (depending on A ) such that if n > N , then s n > A . Proposition 28. lim n →∞ 1 n = 0 lim n →∞ a n = 0 if 0 < a < 1 lim n →∞ a n = if 1 < a Theorem 29. Given two convergent sequences s n and t n , then For any real numbers a and b , then lim n →∞ ( as n + bt n ) = a lim s →∞ s n + b lim s →∞ t n . lim n →∞ ( s n t n ) = lim n →∞ s n lim n →∞ t n . If lim n →∞ t n 6 = 0, then lim n →∞ s n t n = lim n →∞ s n lim n →∞ t n . Theorem 30. Given a sequence of real number s n , If, for any n > N for some natural number N , s n +1 6 s n and if there exists a real M such that s n > M for any n then s n is convergent and lim n →∞ s n > M . If, for any n > N for some natural number N , s n +1 > s n and if there exists a real M such that s n 6 M for any n then s n is convergent and lim n →∞ s n 6 M . 11

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Introductory Real Analysis Math 327, Winter 2017 University of Washington c 2017, Dr. F. Dos Reis Theorem 31. Squeeze theorem Given 3 sequences s n , t n and v n , if there exists a natural number N for any n > N , s n 6 t n 6 v n lim n →∞ s n = lim n →∞ v n = l Then t n is convergent and lim n →∞ t n = l . Theorem 32. Given 2 sequences s n and t n , assume s n 6 t n for any n > N for some N If lim n →∞ s n = then lim n →∞ t n = . If lim n →∞ t n = -∞ , then lim n →∞ s n = -∞ . Theorem 33. Nested Intervals (2.8) Given a sequence of nested closed intervals I n = [ a n , b n ], ( a n 6 a n +1 , and b n > b n + 1), their intersection is either a singleton or a closed interval. 6 Chapter 16 6.1 Open sets, Closed sets Definition 34. A neighborhood of x 0 is a set of points x such that | x - x 0 | < h or equivalently ( x 0 - h, x 0 + h ) for some positive real h . Definition 35. A set S is open if any element s in S has a neigh- borhood ( s - h, s + h ) entirely included in S . A set S is closed if its complementary is open. Remark: the sets • ∅ and ( -∞ , ) are both open and closed. ( a, b ) is open with a being possibly -∞ and b being . R r { a } is open for any real a . [ a, b ], ( -∞ , b ], [ a, ), { a } are closed. The set R r 1 , 1 2 , 1 3 , · · · , 1 n , · · · is neither closed nor open. 12
Introductory Real Analysis Math 327, Winter 2017 University of Washington c 2017, Dr. F. Dos Reis Theorem 36. If A and B are two open sets, then A B is open A B is open.
• Spring '17
• Math, Introductory Real Analysis, Dr. F. Dos Reis, Dr. F. Dos

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