Real Analysis Final

Theorem (Monotonic Subsequences)

Let (a_n)n=1 to inf be any sequence. Then there exists monotonic subsequences whose limits are limsup(a_n) and liminf(a_n).

Defn. (Subsequential Limit)

Let (a_n)n=1 to inf be a sequence. A subsequential limit is any extended real number (can be infinity) that is the limit of a subsequence of (a_n)n=1 to inf. Denoted by S.

Thm (S)

Let (a_n)n=1 to inf be any sequence of real numbers, and S the corresponding set of sequential limits of (a_n)n=1 to inf. Then we have the following:

1) S isn’t the empty set

2) supS=limsup(a_n) and infS=liminf(a_n)

3) lim n-to-inf a_n exists iff S has exactly one element, name the limit of (a_n)n=1 to inf.

Defn. 2.4.3 (Convergence of a Series)

Let (a_n)n=1 to inf be a sequence. An infinite series is a formal expression of the sums of the terms. We define the corresponding sequence of partial sums (s_m)m=1 to inf by

s_m = a_1 + a_2 + … + a_m,

and say the series of sums converges to A if the sequence of partial sums converges to A.

Thm 2.4.6 (Cauchy Condensation Test)

Suppose (a_n) is decreasing and satisfies a_n >= 0 for all n e N. Then, the series converges iff the series

SUM) 2^n*a_2n = a_1 + 2a_2 + 4a_4 + 8a_8 + 16a_16 + …

converges.

Corollary 2.4.7

The series

SUM) 1/n^p

converges iff p > 1.

Thm 2.7.1 (Algebraic Limit Thm for Series)

If the infinite series a_k converges to A and the infinite series b_k converges to B then

1) multiplying the series by a constant c equals cA which is equivalent to pulling the constant c out

2) adding a_k and b_k equals A+B which is equivalent to summing both a_k and b_k seperately.

Thm. 2.7.2 (Cauchy Criterion for Series)

The infinite series of a_k converges iff, given epsilon > 0, there exists an N as an element of the natural numbers such that whenever n>m>=N it follows that

|a_m+1 + a_m+2 + … + a_n| < epsilon.

Said another way, the series a_n converges iff its sequence of partial sums in Cauchy.

Another way, for all epsilon > 0, the exists N as an element of the natural numbers such that for all m,n>=N, |s_n - s_m| < epsilon.

Thm 2.7.3

If the infinite series of a_k converges, then a_n as n approaches infinity converges to 0.

Thm 2.7.4 (Comparison Test)

Assume (a_k)k=1 to inf and (b_k)k=1 to inf are sequences satisfying 0 <= a_k <= b_k for all kinds of. Then

1) if sum(b_k) converges, sum(a_k) converges

2) if sum(a_k) diverges, then sum(b_k) diverges.

Defn 2.7.8 (Absolute/Conditional Convergence)

If SUM(|a_n|) converges, then we say the original series converges absolutely.

If the original series converges but SUM(|a_n|) does not converge, then we say the original series converges conditionally.

Thm 2.7.6 (Absolute Convergence Test)

If the series SUM(|a_n|) converges, then the original series converges as well.

Thm (Root Test)

Let (a_n)n=1 to inf be a sequence, and let A=limsup(|a_n|^1/n). Then the series SUM(a_n)

1) converges absolutely if A<1

2) diverges if A>1

3) test gives no information if A=1

Thm (Ratio Test)

Let (a_n)n=1 to inf be a sequence. Then the series SUM(a_n)

1) coverges absolutely if limsup|a_n+1/a_n|<1

2) diverges if liminf|a_n+1/a_n|>1

3) test gives no information if liminf <= 1 <= limsup

Thm 2.7.7 (Alternating Series Test)

Let (a_n) be a sequence satisfying

1) terms are decreasing

2) (a_n) converges to 0

Then, the alternating series SUM(-1)^n*a_n converges.

Thm (Abel’s Test)

If

1) the infinite series of a_n coverges and

2) (b_n) is a bounded monotone sequence

then the infinite series of a_n*b_n coverges.

Defn 3.2.1 (Open Set)

A set O as a subset of the real numbers is open if for all points a as an element of O there exists an epsilon-neighborhood V_epsilon(a) as a subset of O.

Thm 3.2.3 (Union/Intersection)

1) The union of an arbitrary collection of open sets is open

2) The intersection of a finite collection of open sets is open

Defn 3.2.4 (Limit Point)

A point x is a limit point of a set A if every epsilon neighborhood V_epsilon(x) of x intercepts the set A at some point other than x.

Defn 3.2.6 (Isolated Point)

A point a as an element of A is an isolated point of A if it is not a limit point of A.

Thm 3.2.5 (Limit Points and Sequences)

A point x is a limit point of a set A iff x=lim as n goes to inf of a_n for some sequence (a_n)n=1 to inf contained in A satisfying a_n =/ x for all n as an element of the natural numbers.

Defn 3.2.7 (Closed)

A set F as a subset of the real numbers is closed if it contains its limit points.

Thm 3.2.8 (Closed/Cauchy)

A set F as a subset of the real numbers is closed iff every Cauchy sequence contained in F has a limit that is also an element of F.

Thm 3.2.10 (Density)

For every y as an element of the real numbers, there exists a sequence of rational numbers that converges to y.

Defn 3.2.11 (Closure)

Given a set A as a subset of the real numbers, let L be the set of all limit points of A. The closure of A is defined to be A-bar = A U L

Thm 3.2.12 (Closure)

For any A as a subset of the real numbers, the closure A-bar is a closed set and is the smallest closed set containing A.

Thm 3.2.13 (Open/Closed Relation)

A set O is open iff O^c is closed. Likewise, a set F is closed iff F^c us open.

Thm 3.2.14 (Closed and Union/Intersection)

1) The union of a finite collection of closed sets is closed.

2) The intersection of an arbitrary collection of closed sets is closed.

Defn 3.3.1 (Compactness)

A set K as a subset of the real numbers is compact if every sequence in K has a subsequence that converges to a limit that is also in K.

Defn 3.3.3 (Bounded)

A set A as a subset of the real numbers is bounded if there exists M>0 such that |a|<=M for all a e A.

Thm 3.3.4 (Characterization of Compactness in R)

A set K as a subset of the real numbers is compact iff it is closed and bounded.

Thm 3.3.5 (Nested Compact Set Property)

If

K_1 containing K_2 containing …

is a nested sequence of nonempty compact sets, then its intersection is non-empty.

Defn 3.3.6 (Open Cover)

Let A be a subset of the real numbers. An open cover for A is a (possibly infinite) collection of open sets {O_lambda : lambda is an element of ^} whose union contains the set A.

Given an open cover for A, a finite subcover is a finite subcollection of open sets from the original open cover whose union still manages to completely contain A.

Thm 3.3.8 (Heine-Borel Thm)

Let K be a subset of the real numbers. All of the following statements are equivalent in the sense than any one of them implies the two others:

1) K is compact.

2) K is closed and bounded.

3) Every open cover for K has a finite subcover.

Defn. 3.4.1 (Perfect Set)

A set P as a subset of the real numbers is called perfect if it is closed and contains no isolated points.

Thm 3.4.3 (Perfect/Uncountable)

A non-empty perfect set is uncountable.

Defn 3.4.4 (Separated/Disconnected)

Two non-empty sets A, B as subsets of the real numbers are separated if

A-bar intersect B is the empty set, AND A intersect B-bar is the empty set.

A set E as a subset of the real numbers is called disconnected if it can be written as E = AUB

where A and B are non-empty and separated.

A set that is NOT separated is called connected.

Thm 3.4.6 (Connected/Sequences)

A set as a subset of the real numbers is connected iff for all non-empty disjoint sets A and B such that

E = AUB

there always exists a convergent sequence (x_n)n=1 to inf with (x_n)n=1 to inf contained in one of A or B, and its limit x contained in the other.

Thm 3.4.7 (Connected/Interval and Point)

A set E as a subset of the real numbers is connected iff it is an interval or a single point.

Defn 4.2.1 (Functional Limits)

Let f:A→R, and let c be a limit point of the domain A. We sayt= that

lim x→c f(x) = L

provided that, for all epsilon > 0, there exists a delta >0 such that whenever 0<|x-c|< delta AND x is an element of A it follows that |f(x)-L| < epsilon.

Defn (Functional Limits, Topological)

Let f:A→R and let c be a limit point of the domain A. We say

lim x→c f(x) = L

provided that, for all epsilon neighborhoods V_epsilon(L) of L, there exists a delta neighborhood V_delta( c ) of c such that

f(V_delta( c ) \ {c}) intersect A) is a subset of V_epsilon(L).

Thm 4.2.3 (Sequential Criterion for Functional Limits)

Given a function f:A→R and a limit point c of A, the following two statements are equivalent:

1) lim x→c f(x) = L

2) For all sequences (x_n)n=1 to inf as a subset of A satisfying x_n =/ c and x_n→c, it follows that f(x_n)→L.

Corollary 4.2.4 (Algebraic Limit Thms for Functional Limits)

Let f and g be functions defined on a domain A as a subset of the real numbers, and assume lim f(x) → L and lim g(x) → M for some limit point c of A. Then

1) constant is multiplied through

2) adding functions equals adding limits

3) multiplying functions equals multiplying limits

4) dividing functions equals dividing limits given M=/0.

Corollary 4.2.5 (Divergence Criterion for Functional Limits)

Let f be a function defined on A, and let c be a limit point of A. If there exists two sequences (x_n) and (y_n) in A where they aren’t equal to c, and

lim(x_n)=lim(y_n)=c but lim(f(x_n))=/lim(f(y_n)),

then the limit lim x→c f(x) does not exist.

Defn. 4.3.1 (Continuity)

A function f:A→R is continuous at a point c in A if for all epsilon > 0 there exists a delta > 0 such that whenver |x-c|< delta AND x is in A it follows that |f(x)-f( c)| < epsilon.

If f is continuous at every point in a, f is continuous on A.

Thm 4.3.2 (Characterization of Continuity)

Let f:A→R and let c be in A. f is continuous if any of the 3 are met:

1) thm 4.3.1

2) f(delta neighborhood of c intersect A) is a subset of epsilon neighborhood of f(c )

3) if x_n is an element of A for all n and x_n→c then

f(x_n)→f( c)

Corollary 4.3.3 (Criterion for Discontinuity)

Let f:A→R and let c be a limit point of A. If there exists a sequence (x_n) in A where x_n→c but such that f(x_n) doesn’t converge to f( c), then f is not continous at c.

Thm 4.3.9 (Composition of Continuous Functions)

Given f:A→R and g:B→R, assume that the range f(A)={f(x):x e A) so that the composition g o f(x) = g(f(x)) is defined on A.

If f is continuous at c in A, and if g is continuous at f(c ) in B, the g o f is continuous at c.

Thm 4.4.1 (Preservation of Compact Sets)

Let f:A→R be continuous on A. If K as a subset of A is compact, then f(K) is compact as well.

Thm 4.4.2 (Extreme Value Thm)

If f:K→R is continuous on a compact set K as a subset of R, then f attains a maximum and minimum value. In other words, there exists x_0, x_1 in K such that f(x_0) <= f(x) <= f(x_1) for all x in K.

Defn 4.4.4 (Uniform Continuity)

A function f:A→R is uniformly continuous on A if for every epsilon > 0 there exists a delta > 0 such that for all x, y in A |x-y| < delta implies |f(x)-f(y)| < epsilon

Thm 4.4.7 (Uniform Continuity on Compact Sets)

A function that is continuous on a compact set K is uniformly continuous on K.

Thm 4.5.1 (Intermediate Value Thm)

Let f:A→R be continuous. If L is a real number satisfying f(a)<L<f(b) or f(a)>L>f(b), then there exists a point c e (a,b) where f(c ) = L.

Thm 5.2.3 (Differentiable/Continuous)

If f:A→R is differentiable at a point c in A, then f is continuous at c as well.

Thm 5.3.1 (Rolle’s Thm)

Let f:[a,b]→R be continuous on [a,b] and differentiable on (a,b). If f(a)=f(b), then there exists a point c in (a,b) where f’(c ) = 0.

Thm 5.3.2 (Mean Value Thm)

If f:[a,b]→R is continuous on [a,b] and differentiable on (a,b), then there exists a point c e (a,b) where

f(b)-f(a)=f’(c )(b-a)