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.

### Present Remotely

Send the link below via email or IM

CopyPresent to your audience

Start remote presentation- Invited audience members
**will follow you**as you navigate and present - People invited to a presentation
**do not need a Prezi account** - This link expires
**10 minutes**after you close the presentation - A maximum of
**30 users**can follow your presentation - Learn more about this feature in our knowledge base article

Do you really want to delete this prezi?

Neither you, nor the coeditors you shared it with will be able to recover it again.

### Make your likes visible on Facebook?

Connect your Facebook account to Prezi and let your likes appear on your timeline.

You can change this under Settings & Account at any time.

# Real Analysis Review

No description

by

Tweet