real analysis exam 2 flashcards

Metric (definition)

A function d: X×X → ℝ is a metric if for all p,q,r:

1. d(p,q) ≥ 0

2. d(p,q) = 0 ⇔ p = q

3. d(p,q) = d(q,p)

4. d(p,r) ≤ d(p,q) + d(q,r)

Open ball

B(p,r) = {x ∈ X : d(x,p) < r}

Open set

U ⊆ X is open if every x ∈ U has r>0 with B(x,r) ⊆ U.

Closed set

F ⊆ X is closed if it contains all its limit points.

Equivalent: X \ F is open.

Interior

int(A) = {x ∈ A : ∃r>0, B(x,r) ⊆ A}

Closure

Ā = A ∪ {limit points of A}

Characterizations

A is open ⇔ A = int(A)

A is closed ⇔ A = Ā

Arbitrary union of open sets is open

If {Uα} are open, then ⋃Uα is open.

Finite intersection of open sets is open

Infinite union of closed sets may not be closed

Example: ⋃[0,1−1/n] = [0,1)

Boundary

∂E = Ē ∩ (Eᶜ)̄

E is open ⇔ E ∩ ∂E = ∅

Limit of a sequence

xₙ → L means:

∀ε>0 ∃N ∀n≥N, d(xₙ, L) < ε.

Bounded sequence

∃M>0, p ∈ X such that d(xₙ,p) ≤ M for all n.

Limit laws

If xₙ→a and yₙ→b:

- xₙ + yₙ → a + b

- cxₙ → ca

- If xₙ ≤ yₙ ∀n, then a ≤ b

Cesàro mean

If xₙ → L, then (x₁+...+xₙ)/n → L.

Interleaving convergence

xₙ → L ⇔ (x₁, L, x₂, L, x₃, L, ...) converges.

Every real sequence has a monotone subsequence

Cauchy sequence

∀ε>0 ∃N ∀m,n≥N, d(xₙ,xₘ) < ε.

Complete metric space

Every Cauchy sequence converges in the space.

Closed subset of a complete space is complete

If X is complete and E ⊆ X is closed, then any Cauchy sequence in E converges to a point in E.

Sequence limit implies closedness

If xₙ → p, then {p, x₁, x₂, ...} is closed.

Compact set

Every open cover has a finite subcover.

Heine-Borel (ℝⁿ)

K ⊆ ℝⁿ is compact ⇔ closed and bounded.

Finite union of compact sets is compact

Infinite union of compact sets may not be compact

Example: ⋃[0,1−1/n] = [0,1)

ℝ is not compact

Not bounded; open cover {(-n,n)} has no finite subcover.

Usual metric on R

d(x,y)=|x−y|. Standard distance on the real line.

Taxicab (Manhattan) metric

d(x,y)=|x1−y1|+|x2−y2|+...+|xn−yn|. Grid-based distance.

ℓ¹ metric on sequences

d(x,y)=Σ(|xn−yn|/2^n).

Usual metric on R^n

d(x,y)=√((x1−y1)^2+...+(xn−yn)^2).

Discrete metric

d(x,y)=0 if x=y, and 1 if x≠y.

Open interval (a,b) is open

Every point x in (a,b) has a small radius r so that (x−r, x+r) stays inside (a,b).

Closed interval [a,b] is closed

The limit of any convergent sequence in [a,b] stays in [a,b], so it contains all its limit points.

Interior of a set

The interior int(A) is the set of all points of A that have an open ball fully contained in A.

Closure of a set

The closure Ā is A plus all its limit points; smallest closed set containing A.

Boundary of a set

The boundary ∂A = Ā ∩ (Aᶜ)̄; points where every ball intersects both A and its complement.

A set is open iff A ∩ ∂A = ∅

Open sets contain no boundary points; every boundary point touches the outside.

A set is closed iff it contains all its limit points

If every convergent sequence in A has its limit in A, then A is closed.

Bounded monotone sequence theorem

Every bounded monotone sequence converges.

Every sequence has a monotone subsequence

Any real sequence contains a monotone subsequence (key lemma for Bolzano-Weierstrass).

Limit of monotone increasing sequence

If xₙ is increasing and convergent, its limit equals sup{xₙ}.

Limit of monotone decreasing sequence

If xₙ is decreasing and convergent, its limit equals inf{xₙ}.

Balzano wienerstrass theorem

Let xn be a sequence contained in a compact metric space k, the xn has a convergent subsequence.

Accumulation point

If E contained in X , then p in x is a accumulation point of E if for all ɛ>0 we have B’(ɛ,p) intersection E ≠ the empty set where B’ is a ball with no center