R eal nalysis A {Sets} Metric Theorems C omplete auchy ompact onnected onvergence Real axioms Open closed sequence series An open ball of radius r around x is {a|d(x,a)<r}. An open set is the union of open balls. A closed set contains its limit points. A set can be open and closed at the same time. This is called 'clopen'.

A set is complete if every limit point is in the set. R is complete, Q is not. A sequence is Cauchy if for any ε there is an N such that for all n>N, x-x<ε n n-1 In a Complete space, every Cauchy sequence Converges A sequence (a) converges to a limit x if for any ε there exists N such that for all n>N, d(a,x)<ε n n A sequence (a) is an ordered infinite set of numbers. A series is the sum of a sequence. n If for any cover of your set you have a finite subcover, your set is compact. In a complete metric space, a set is compact iff it is closed and bounded. A set is disconnected if it can be written as the disjoint union of two non-empty clopen sets. A set is connected if it is not disconnected. R is a field. + and * are commutative and associative. 0 is the + identity, 1 is the * + and * inverses exist. * distributes over +.

x*0=0

-x=-1*x

a=/=-a unless a=0

R is infinite

For all positive a, x =a exists. 2 LUB A set has the least upper bound property if every set that is bounded has a lowest upper bound. R has LUB. If f is a real valued function from [a,b] to R and f(a)<γ<f(b), then there exists a<c<b such that f(c)=γ IVT A function d such that d(a,b)≥0, d(a,b)=0 iff a=b, d(a,b)=d(b,a), and d(a,c)≤d(a,b)+d(b,c) is a distance function, or metric. The standard Euclidean metric is d((a,b,c,...),(α,β,ξ,...))=((a-α)+(b-β)+(c-ξ) ) 2 2 2 1/2 |<x,y>|≤|x||y| Cauchy Schwarz Every bounded sequence of real numbers has a convergent subsequence BWT Every sequence of real numbers has a monotone subsequence: select all the peaks, or if there are finitely many of them select all the troughs. Any bounded monotone sequence is convergent. TFAE f is continuous at a. For every ε>0 there exists δ>0 such that if d(x,a)<δ then d(f(x),f(a)<ε Given any sequence (x) converging to a, (f(x)) converges to f(a) n n If f is continuous, then the limit of f(a) approaches f(x) as a approaches x. Thus (1) implies (2). Select all (x) from (2) and that's your sequence converging to a, and if d(f(x),f(a))<ε, that implies f(x) converges to f(a), thus (2) implies (3). Theorem 4.2.4 says (3) implies (1). TFAE Let S be a subset of X, a metric space. S has finite diameter. There exist a in X and r>0 Given any point a in X there exists r>0 such that S is a subset of a ball of radius r centered at a.

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# Real Analysis Review

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