Math 521 Midterm 2

sequence convergence

x_n converges to x∈R if ∀ε>0, ∃N∈N s.t. n≥N |x_n-x|<ε

showing that x_n converges to x; ex- show 1/n converges to 0

|(1/n)-0|=1/n; as n≥N, (1/n)≤(1/N)<ε. Proof: Let ε>0. By AP ∃N∈N s.t. N>(1/ε) aka (1/N)<ε. Let n≥N, then |(1/n)-0|=(1/n)≤(1/N)<ε. Thus 1/n converges to 0

sequence bounded

a sequence x_n is bdd iff ∃M>0 s.t. ∀n∈N, |x_n|≤M

Algebraic Limit Theorem

suppose x_n and y_n are sequences s.t. x_n→x and y_n→y. (x_n+y_n)→(x+y); ∀c∈R, cx_n→cx; x_ny_n→xy; if x_n≠0 ∀n∈N and x≠0 then (1/x_n)=(1/x)

order limit theorem

if x_n≥y_n ∀n∈N and x_n→x and y_n→y then x≥y

squeeze theorem

let x_n, y_n and z_n be sequences. suppose ∃N₀∈N s.t. n≥N₀ and x_n≤y_n≤z_n and x_n→w and z_n→w then y_n→w

binomial theorem

(1+x)ⁿ=1 + nx + (nx²(n-1))/2! + (nx³(n-1)(n-2))/3! + …

bernoulli inequality

(1+x)ⁿ≥(1+nx)

theorem with |b|

if |b|<1 then bⁿ→0

abs value sequence convergence

|a_n|→0 iff a_n→0

monotone sequence

x_n is monotone if x_n≤x_n+1 or x_n+1≤x_n

monotone convergence theorem

let x_n be a monotone sequence of real numbers. x_n converges iff x_n is bounded

subsequence def

if x_n is a sequence and n_k is a strictly increasing sequence of natural numbers, then x_nk is a subsequence of x_n

well-ordering property of N

if n₁<n₂<n₃<… is a strictly increasing sequence of natural numbers then ∀k∈N, n_k≥k

convergence and subsequence theorem

if x_n→x then every subsequence x_nk→x

cauchy

A sequence x_n is cauchy if ∀ε>0 ∃N∈N s.t. ∀n,m ≥ N, |x_n-x_m|<ε

open neighborhood

Let a∈R and ε>0. The set V_ε(a)={x∈R : |x-a|<ε}=(a-ε,a+ε) is an open neighborhood of a of radius ε

limit point (neighborhood)

x∈R is a limit point of set E if ∀ε>0 ∃q∈E q≠x s.t. q∈V_ε(x) (AKA |q-x|<ε)

limit point (sequence)

x is a limit point of set E IFF ∃x_n with x_n∈(E\{x}) ∀n s.t. x_n→x

isolated point def

if p∈E and p is not a limit point of E, then p is an isolated point

closed set

a set E is closed if every limit point of E is an element of E

interior point

p is an interior point of E if ∃ε>0 s.t. V_ε(p)⊆E

open set

a set E is open if every point in E is an interior point

complement open/closed

E is open IFF E^c is closed; F is closed IFF F^c is open

{G_a} open sets

U(G_a) is open

{F_a} closed sets

∩(F_a) is closed

G₁…G_N

if G₁…G_N are open then ∩(G_j) is open (j=1 to N)

F₁…F_N

if F₁…F_N are closed then U(F_j) is closed (j=1 to N)

E closure is

E U E’ = E union (set of limit points of E)

in E closure, theorem

p∈E_closure IFF ∃x_n of elements in E s.t. x_n→p

theorem, let E⊆R (closure stuff)

E_closure is closed; E=E_closure IFF E is closed; if F is closed and E⊆F then E_closure⊆F; hold for all E⊆R, no need to prove one of them first

open AND closed sets

E is open and closed IFF E=∅ or E=R

open cover of E

a collection of open sets {G_a} s.t. E⊆U(G_a)

compact def

a set K is compact if every open cover of K has a finite subcover; If {G_a} is an open cover of K, then then ∃a₁,a₂,…a_N s.t. K⊆U(G_aj) (j=1 to N)

Let E⊆R, the following are equivalent (compactness stuff)

E is compact; E is closed and bounded; Every infinite subset of E has a limit point in E; Every sequence in E has a subsequence that converges to an element in E; one of the statements must hold and then the rest hold

closed subsets of compact sets theorem

closed subsets of compact sets are compact

Bolanzo-Weierstrass Theorem

if x_n is a bounded sequence then it has a convergent subsequence

Cauchy criterion / Completeness of R

x_n converges IFF x_n is a cauchy sequence