such that for all n N s n s This is by definition equivalent to s n N s Since

# Such that for all n n s n s this is by definition

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such that for all n N, | s n - s | < . This is by definition equivalent to s n N ( s ) . Since by hypotheses s n F for all n, and moreover we have assumed that s is not in F, consequently s n N * ( s ) F. We have just shown that s satisfies the definition of being an accumulation point of F. (why?) Hence, s F, by Theorem 13.17.
MATH 117 LECTURE NOTES FEBRUARY 17, 2009 5 (5) This is Theorem 16.13 in the text. Assume { s n } n =1 is a convergent sequence of real numbers and its limit is s. Then, by definition of convergence (geez, go on and memorize it already!) using = 1 , there exists N N such that for all n N, | s n - s | < 1 ⇒ | s n | < | s | + 1 for all n N. So define M = | s 1 | + | s 2 | + . . . + | s N - 1 | + | s | + 1 . For any n N, | s n | < | s | + 1 < M. For any n N - 1 , | s n | < M. So the sequence is bounded by M.
6 DR. JULIE ROWLETT (6) This was done in class on Tuesday February 10, 2009. The set of numbers { s n } n =1 is bounded above by s 1 and below by 0 . Let s be the infimum of S = { s n } n =1 . Then, let > 0 . By definition of infimum (did you memorize this?), s + is no longer a lower bound for S. Hence, there is some s N S such that s N < s + . However, the sequence is decreasing, so for all n N, s n s N < s + . By definition of infimum, for all n N , s s n . Hence, for all n N, s s n s + ⇒ | s n - s | < . This is the definition of convergence.
MATH 117 LECTURE NOTES FEBRUARY 17, 2009 7 2. Monotone Sequences Definition 2.1. A sequence { s n } n =1 is increasing if s n s n +1 for all n N . The sequence is decreasing if s n s n +1 for all n N . A sequence which is either increasing or decreasing (by the above definitions) is called monotone.

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• Fall '08
• Akhmedov,A
• Math, Order theory, Sn, Dominated convergence theorem, Monotone convergence theorem, DR. JULIE ROWLETT

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