# Analysis Final

- StudyBlue
- Wisconsin
- University of Wisconsin - Madison
- Mathematics
- Mathematics 521
- Stovall
- Analysis Final

**Created:**2016-12-17

**Last Modified:**2016-12-22

1) (+ closure), + commutes, + associative, + identity, + inverse

2) multiplicative closure, commute, associative, identity, inverse, distributivity

1) for any x,y in S, exactly one of the following holds: x=y, x<y, y<x

2) Transitivity: for any x, y, z in S, if x < y and

y<z, then x <z

1) if y is less than or equal to z, then x + y is less than or equal to x +z

2) if x is greater than or equal to 0 and y is greater than or equal to 0, then x*y is greater than or equal to 0.

i) (positivity) d(x,y) is greater than or equal to 0 and d(x,y) = 0 if and only if x = y

ii) (symmetry) d(x,y) = d(y,x)

iii) (Triangle Inequality) d(x,z) is less than or equal to d(x,y) + d(y,z)

_{r}(p) is contained in E.

_{r}(p) ∩ E) \ {p} ≠ ∅

_{r}(p) ∩ E contains infinitely many points.

2) arbitrary intersections and finite unions of closed sets are closed

^{k}with Euclidean metric, a set is compact <=> it is closed and bounded.

A ≠ ∅, B ≠ ∅, Ā ∩ B = ∅ and

B̄ ∩ A = ∅

Ā ∩ B = ∅ and B̄ ∩ A = ∅

E Is not connected if there exists a separation of E

_{n}) in X converges to p( in X) if for any Ɛ > 0 there exists N in N(natural numbers) such that for any n > N, d(p

_{n},p) < Ɛ.

_{n}), (y

_{n}), (z

_{n}) be three sequences in R. Assume that Lim x

_{n}= x = lim z

_{n}and that for any n in N (natural numbers) x

_{n}≤ y

_{n}≤ z

_{n}. Then lim y

_{n}= x.

_{n}) if there exists a subsequent (p

_{nk}) of (p

_{n}) such that limit as k goes to infinity of p

_{nk}= p.

_{n}) be a Sequence in X. Then p in X is a sub-sequential limit of (p

_{n}) <=> for any Ɛ > 0, N

_{Ɛ}(p) contains p

_{n}for infinitely many n's.

_{n}) is a Cauchy Sequence if for any Ɛ > 0 there exists an N in N (natural numbers) such that for any n, m ≥ N , d(p

_{n}, p

_{m}) < Ɛ.

_{n}in R is increasing if S

_{n+1}≥ S

_{n}for any n in N(natural numbers)

_{n}) in R is decreasing if S

_{n+1}≤ S

_{n}for any n in N(natural numbers)

_{n}is monotone if ( for any n in N (natural numbers) S

_{n+1}≥ S

_{n}) or ( for any n in N(natural numbers) S

_{n+1}≤ S

_{n}.

_{n}) is not monotone if there exists an n in N(natural numbers) such that S

_{n+1}< S

_{n}and if there exists an m in N(natural numbers) such that S

_{m+1}> S

_{m}.

_{n}) be a Sequence in R. Say (S

_{n}) diverges to +∞ if for any M > 0 there exists an N in N (natural numbers) such that for any n ≥ N, S

_{n}> M

_{n}converges absolutely if the summation of |a

_{n}| converges.

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