Math 341 final exam review

infinite series

∑bn = b1 + b2 + b3 + ... it converges if it's sequence of partial sums converges to B

Continuous Function

(i) For all ε> 0, there exists a δ > 0 such that |x−c| < δ (and x ∈ A) implies |f(x)−f(c)| <ε ; (ii) limx→c f(x) = f(c); (iii) For all Vε(f(c)), there exists a Vδ(c) with the property that x ∈ Vδ(c) (and x ∈ A) implies f(x) ∈ Vε(f(c)); (iv) If (xn) → c (with xn ∈ A), then f(xn) → f(c).

least upper bound

(i) s is an upper bound for A; (ii) if b is any upper bound for A, then s ≤ b

Partition

A _______P of [a, b] is a finite, ordered set P = {a = x0 < x1 < x2 < ··· < xn = b}. For each subinterval [xk−1, xk] of P, let mk = inf{f(x) : x ∈ [xk−1, xk]} and Mk = sup{f(x) : x ∈ [xk−1, xk]}.

Increasing (decreasing) function

A function f : A → R is ______on A if f(x) ≤ f(y) whenever x<y and _______ if f(x) ≥ f(y) whenever x<y in A.

Intermediate value property

A function f has the _________ on an interval [a,b] if for all x < y and all L between f(x) and f(y), it is always possible to find a point c ∈ (x,y) where f(c) = L.

Bounded

A set A ⊆ R is ______ if there exists M > 0 such that |a| ≤ M for all a ∈ A.

bounded above

A set A ⊆ R is _________ if there exists a number b ∈ R such that a ≤ b for all a ∈ A. The number b is called an________ for A

Disconnected sets

A set E ⊆ R is ________ if it can be written as E = A∪B, where A and B are nonempty separated sets.

Compact

A set K ⊆ R is _____ if every sequence in K has a sub-sequence that converges to a limit that is also in K.

open sets

A set O ⊆ R is ____ if for all points a ∈ O there exists an epsilon-neighborhood V(a) ⊆ O

Perfect Set

A set P ⊆ R is ______ if it is closed and contains no isolated points.

Open Cover

An _______ for A is a (possibly infinite) collection of open sets whose union contains the set A.

closure

Given a set A ⊆ R, let L be the set of all limit points of A. The______ of A is defined to be A = A∪L.

Uniform Continuity

If neither if there exists a particular ε0 > 0 and two sequences (xn) and (yn) in A satisfying |xn −yn| → 0 but |f(xn)−f(yn)| ≥ ε0.

Lower integral

L(f) = sup{U(f,P) : P ∈ P}

Lower Sum

L(f,P) = from k=1 to n ∑mk(xk −xk−1).

Limit of a function

Let f : A → R, and let c be a limit point of the domain A. We say that limx→c f(x) = L provided that, for all ε> 0, there exists a δ > 0 such that whenever 0 < |x−c| < δ (and x ∈ A) it follows that |f(x)−L| < ε.

Isolated Point

Not a limit point

Connected sets

Not disconnected

Refinement

P if Q contains all of the points of P. In this case, we write P ⊆ Q.

Separated Sets

Two nonempty sets A, B ⊆ R are ______ if A' ∩ B and A ∩ B' are both empty.

Upper integral

U(f) = inf{U(f,P) : P ∈ P}

upper sum

U(f,P) =from k=1 to n ∑Mk(xk −xk−1)

sequence

domain is N

axiom of completeness

every nonempty set of real numbers that is bounded above has a least upper bound

convergence of a sequence

for every positive number , there exists an N ∈ N such that whenever n ≥ N it follows that |an −a| < ε.

Cauchy Criterion

for every ε > 0, there exists an N ∈ N such that whenever m, n ≥ N it follows that |an −am| < ε.

Riemann Integrability

from a to b ∫ f=U(f) = L(f).

countable

if N~A (same cardinality)

closed sets

it contains all limit points

Limit point

of a set A if and only if x = liman for some sequence (an) contained in A satisfying an = x for all n ∈ N.

Monotone sequence

sequence that is neither increasing nor decreasing