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Unformatted text preview: Math 117, F 09 Review Outline for First Midterm Deﬁnitions.
sequence
converge
neighborhood
bounded from above/below (a sequence or a set)
bounded (a sequence or a set)
convergent/divergent
limit (of a convergent sequence)
Cauchy sequence
accumulation point
lower/upper bound
greatest lower bound / least upper bound
inﬁmum/supremum
Least Upper Bound Property of R
subsequence
increasing/decreasing sequence
monotone sequence
Symbols.
N, Z, Q, R
{a n }∞
n=1
sup / inf
Theorems. This is a list of the major theorems (and corollaries, etc.) that we have
developed so far, with phrases labelling them. You don’t need to memorize the numbers
of these theorems, just the statements. The numbers are just listed here to help you ﬁnd
things in the book.
(Lemma, p.35) convergence in terms of neighborhoods
(1.1) uniqueness of limits of sequences
(1.2) convergent implies bounded
(1.3) convergent implies Cauchy
(1.4) Cauchy implies bounded
(Lemma, p.39) alternate condition for an accumulation point
(0.20) existence of greatest lower bounds
(1.6) The BolzanoWeierstrass Theorem
(1.7) Cauchy implies convergent
(1.8) sum of two convergent sequences
(1.9) product of two convergent sequences (1.10)
(1.11)
(1.12)
(1.14)
(1.15)
(1.16) bounding a sequence away from zero
quotient of two convergent sequences
inequalities between two convergent sequences
convergence via subsequences
convergence of bounded sequences
convergence of monotone sequences ...
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 Fall '08
 Akhmedov,A
 Math

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