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7 Cards in this Set
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Limit

Let f:D R be a function with D in R, xo in R is an accupoint of D. Moreover, let L be in R. We say f(s) has a limit L at xo iff the following holds:
For every positive real number E, there exists a positive real d, such that whenever x is in D\{xo}, with xxo<d, we have f(x)L<E. 

Subsequence

Let (an)(n in N) be a sequence and (nk)(k in N) be a sequence of positive integers such that n1<n2<n3<…<nk. The sequence (nk) is called a subsequence of (an).
In other words: deleting some or none of the terms from a sequence and renaming the remaining terms, with remaining terms in the same order as in the sequence. There are still infinitely many terms after some are deleted. 

Increasing

sequence (an)(n in N) is increasing iff an<=an+1 for all positive integers


decreasing

Decreasing: a sequence is decreasing iff an>=an+1 for all positive integers n.


Monotone

a sequence is either increasing or decreasing


Summary of proofs with lim arithmetic

Take arbitrary sequence that converges to x0
Use thm 16lim(fxn)=A Use some other rule for lim seq Use thm 16 in other direction 

Summary of proofs with convergent arithmetic

Set up convergent definition of an and bn with N1 and N2
(For multiplication add an<=M bounded) (For division bn>c by lemma) Define E 1. For additionE=E/2 2. For multipleE’=E/(/(B+M)>0 3. For divisionE=BE*c N=max(N1,N2) and then put everything together Will use triangle inequality 