MAT371defs1

# MAT371defs1 - . (b) Every neighborhood of x contains...

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Chapter 1 1. Sequence 2. Convergence of a sequence : A sequence { a n } n =1 is said to converge to the real number A iﬀ for ² > 0 there is a positive integer N such that for all n N we have | a n - A | < ² . 3. If ² > 0 the the interval ( x - ²,x + ² ) is called an epsilon neighborhood of x . 4. A set Q of real numbers is said to be a neighborhood of x if there exists an ² > 0 such that ( x - ²,x + ² ) Q. 5. A sequence { a n } n =1 is said to be Cauchy iﬀ for ² > 0 there is a positive integer N such that for all n N and all m N we have | a n - a m | < ² . 6. Let S be a set of real numbers. A real number A is said to be an accumulation point of S iﬀ every neighborhood of A contains inﬁnitely many points of S . 7. Equivalent statements for the deﬁnition of an accumulation point: (a) Every epsilon neighborhood of x contains inﬁnitely many points of S
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Unformatted text preview: . (b) Every neighborhood of x contains innitely many points of S . (c) Every neighborhood of x contains a point of S that is dierent from x. (d) Every epsilon neighborhood of x contains a point of S that is dierent from x. 8. Bolzano-Weierstrass Theorem : Every bounded innite set of real numbers has at least one accu-mulation point. 9. Theorem : Every Cauchy sequence converges to real number. (This is equivalent to the completeness property of real numbers). 10. Subsequence . 11. Theorem Every bounded sequence has a convergent subsequence. 12. Increasing sequence. Decreasing sequence. Monotone sequence . 13. Theorem : Every bounded monotone sequence is convergent....
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## This note was uploaded on 08/12/2009 for the course MAT 371 taught by Professor Thieme during the Spring '07 term at ASU.

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