Math 412 - Final

Cumulative with Chapter 5 material

Axiom of Completeness

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

bounded above/bounded below

A set A ⊆ R is bounded above if there exists a number b ∈ R

such that a ≤ b for all a ∈ A. The number b is called an upper bound for A.

Similarly, the set A is bounded below if there exists a lower bound l ∈ R satisfying l ≤ a for every a ∈ A.

least upper bound (supremum)

A real number s is the least upper bound for a set A ⊆ R if it meets the following two criteria:

(i) s is an upper bound for A;

(ii) if b is any upper bound for A, then s ≤ b.

greatest lower bound (infimum)

A real number l is the greatest lower bound for a set A ⊆ R if

it meets the following two criteria:

(i) l is a lower bound for A;

(ii) if b is any lower bound for A, then b ≤ l.

maximum/minimum

A real number a₀ is a maximum of the set A if a₀ is an element of A and a₀ ≥ a for all a ∈ A. Similarly, a number a₁ is a minimum of A if a₁ ∈ A and a₁ ≤ a for every a ∈ A.

Assume s ∈ R is an upper bound for a set A ⊆ R. Then, s = sup A if and only if…

… for every choice of ε > 0, there exists an element a ∈ A

satisfying s − ε < a.

Nested Interval Property

For each n ∈ N, assume we are given a closed interval I_n = [a_n, b_n] = {x ∈ R : a_n ≤ x ≤ b_n}. Assume also that each I_n contains I_n+1. Then, the resulting nested sequence of closed intervals

I₁ ⊇ I₂ ⊇ I₃ ⊇ I₄ ⊇ · · ·

has a nonempty intersection; that is, ∩^∞_n=1 I_n ≠ ∅.

Archimedean Property

(i) Given any number x ∈ R, there exists an n ∈ N satisfying n > x.

(ii) Given any real number y > 0, there exists an n ∈ N satisfying 1/n < y.

Density of Q in R

For every two real numbers a and b with a < b, there exists a rational number r satisfying a < r < b.

Given any two real numbers a < b…

… there exists an irrational number t satisfying a < t < b.

There exists a real number α ∈ R…

… satisfying α² = 2.

sequence

A sequence is a function whose domain is N.

Convergence of a Sequence

A sequence (a_n) converges to a real number a if, for every positive number ε, there exists an N ∈ N such that whenever n ≥ N it follows that |a_n − a| < ε.

ε-neighborhood

Given a real number a ∈ R and a positive number ε > 0, the set

V(a) = {x ∈ R : |x − a| < ε}

is called the ε-neighborhood of a.

Convergence of a Sequence: Topological Version

A sequence (a_n) converges to a if, given any ε-neighborhood V_ε(a) of a, there exists a point in the sequence after which all of the terms are in V_ε(a). In other words, every ε-neighborhood contains all but a finite number of the terms of (a_n).

Template for a proof that (x_n) → x

- “Let ε > 0 be arbitrary.”

- Demonstrate a choice for N ∈ N. This step usually requires the most

work, almost all of which is done prior to actually writing the formal

proof.

- Now, show that N actually works.

- “Assume n ≥ N.”

- With N well chosen, it should be possible to derive the inequality

|x_n − x| < ε.

Uniqueness of Limits

The limit of a sequence, when it exists, must be unique.

divergence

A sequence that does not converge is said to diverge.

bounded

A sequence (x_n) is bounded if there exists a number M > 0 such that |x_n| ≤ M for all n ∈ N.

Every convergent sequence… (bounds)

… is bounded.

Algebraic Limit Theorem

Let lim a_n = a, and lim b_n = b. Then,

(i) lim(ca_n) = ca, for all c ∈ R;

(ii) lim(a_n + b_n) = a + b;

(iii) lim(a_nb_n) = ab;

(iv) lim(a_n/b_n) = a/b, provided b ≠ 0.

Order Limit Theorem

Assume lim a_n = a and lim b_n = b.

(i) If a_n ≥ 0 for all n ∈ N, then a ≥ 0.

(ii) If a_n ≤ b_n for all n ∈ N, then a ≤ b.

(iii) If there exists c ∈ R for which c ≤ b_n for all n ∈ N, then c ≤ b. Similarly, if a_n ≤ c for all n ∈ N, then a ≤ c.

monotone

A sequence (a_n) is increasing if a_n ≤ a_n+1 for all n ∈ N and

decreasing if a_n ≥ a_n+1 for all n ∈ N. A sequence is monotone if it is either

increasing or decreasing.

Monotone Convergence Theorem

If a sequence is monotone and bounded, then it converges.

Convergence of a Series

Let (bn) be a sequence. An infinite series is a formal expression of the form

Σ^∞_{n =1} b_n = b₁ + b₂ + b₃ + b₄ + b₅ + · · · .

We define the corresponding sequence of partial sums (s_m) by

s_m = b₁ + b₂ + b₃ + · · · + b_m,

and say that the series Σ^∞_{n =1} b_n converges to B if the sequence (s_m) converges to B. In this case, we write Σ^∞_{n =1} b_n = B.

Cauchy Condensation Test

Suppose (b_n) is decreasing and satisfies b_n ≥ 0 for all n ∈ N. Then, the series Σ^∞_{n =1} b_n converges if and only if the series

Σ^∞_{n = 0} 2ⁿb_2ⁿ = b₁ + 2b₂ + 4b₄ + 8b₈ + 16b₁₆ + · · ·

converges.

subsequence

Let (a_n) be a sequence of real numbers, and let n₁ < n₂ < n₃ < n₄ < n₅ < . . . be an increasing sequence of natural numbers. Then the

sequence

(a_n1, a_n2, a_n3, a_n4, a_n5, . . .)

is called a subsequence of (a_n) and is denoted by (a_nk), where k ∈ N indexes the subsequence.

Subsequences of a convergent sequence…

… converge to the same limit as the original sequence.

Bolzano–Weierstrass Theorem

Every bounded sequence contains a convergent subsequence.

Cauchy sequence

A sequence (a_n) is called a Cauchy sequence if, for every ε > 0, there exists an N ∈ N such that whenever m, n ≥ N it follows that |a_n − a_m| < ε.

Every convergent sequence… (Cauchy)

… is a Cauchy sequence.

Cauchy sequences are…

… bounded.

Cauchy Criterion

A sequence converges if and only if it is a Cauchy sequence.

Algebraic Limit Theorem for Series

If Σ^∞_k=1 a_k = A and Σ^∞_k=1 b_k = B, then

(i) Σ^∞_k=1 ca_k = cA for all c ∈ R and

(ii) Σ^∞_k=1(a_k + b_k) = A + B.

Cauchy Criterion for Series

The series Σ^∞_k=1 a_k converges if and only if, given ε > 0, there exists an N ∈ N such that whenever n > m ≥ N it follows that

|a_m+1 + a_m+2 + · · · + a_n| < ε.

If the series Σ^∞_k=1 a_k converges…

… then (a_k) → 0.

Comparison Test

Assume (a_k) and (b_k) are sequences satisfying 0 ≤ a_k ≤ b_k for all k ∈ N.

(i) If Σ^∞_k=1 b_k converges, then Σ^∞_k=1 a_k converges.

(ii) If Σ^∞_k=1 a_k diverges, then Σ^∞_k=1 b_k diverges.

Absolute Convergence Test

If the series Σ^∞_n=1 |a_n| converges, then Σ^∞_n=1 a_n converges as well.

Alternating Series Test

Let (a_n) be a sequence satisfying,

(i) a₁ ≥ a₂ ≥ a₃ ≥ · · · ≥ a_n ≥ a_n+1 ≥ · · · and

(ii) (a_n) → 0.

Then, the alternating series Σ^∞_n=1 (−1)ⁿ⁺¹a_n converges.

converges absolutely/conditionally

If Σ^∞_n=1 |a_n| converges, then we say that the original series Σ^∞_n=1 a_n converges absolutely. If, on the other hand, the series Σ^∞_n=1 a_n converges but the series of absolute values Σ^∞_n=1 |a_n| does not converge, then we say that the original series Σ^∞_n=1 a_n converges conditionally.

rearrangement

Let Σ^∞_k=1 a_k be a series. A series Σ^∞_k=1 b_k is called a rearrangement of Σ^∞_k=1 a_k if there exists a one-to-one, onto function f : N → N such that b_f(k) = a_k for all k ∈ N.

If a series converges absolutely…

… then any rearrangement of this series converges to the same limit.

Cantor set

C = [0,1] \ [(1/3,2/3) ∪ (1/9,2/9) ∪ (7/9,8/9) ∪ …]. Cn = ∩^∞_n=0 C_n . Note that C_n consists of 2ⁿ closed intervals of length 1/3ⁿ , while C itself has length 0 and is closed and perfect.

Open set

A set O ⊆ R is open if for all points a ∈ O there exists an ε-neighborhood V_ε(a) ⊆ O.

Unions and intersections of open sets

(i) The union of an arbitrary collection of open sets is open.

(ii) The intersection of a finite collection of open sets is open.

Limit point

A point x is a limit point of a set A if every ε-neighborhood V_ε(x) of x intersects the set A at some point other than x.

Limit point (sequential def)

A point x is a limit point of a set A if and only if x = lima_n for some sequence (a_n) contained in A satisfying a_n ≠ x for all n ∈ N.

Isolated point

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

Closed set

A set F ⊆ R is closed if it contains its limit points.

Closed sets (Cauchy sequence def)

A set F ⊆ R is closed if and only if every Cauchy sequence contained in F has a limit that is also an element of F .

Density of Q in R (sequential def)

For every y ∈ R, there exists a sequence of rational numbers that converges to y.

Closure

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

Properties of the closure

For any A ⊆ R, the closure A is a closed set and is the smallest closed set containing A.

Relationship between open sets and closed sets (complements)

A set O is open if and only if O^c is closed. Likewise, a set F is closed if and only if F^c is open.

Unions and intersections of closed sets

(i) The union of a finite collection of closed sets is closed.

(ii) The intersection of an arbitrary collection of closed sets is closed.

Compact sets

A set K ⊆ R is compact if every sequence in K has a subsequence that converges to a limit that is also in K.

Bounded sets

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

Heine Borel Theorem

A set K ⊆ R is compact if and only if it is closed and bounded.

Nested Compact Set Property

If

K₁ ⊇ K₂ ⊇ K₃ ⊇ K₄ ⊇ · · ·

is a nested sequence of nonempty compact sets, then the intersection ∩^∞_n=1 K_n is not empty.

Perfect set

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

Are perfect sets countable?

A nonempty perfect set is uncountable.

Connected set

Two nonempty sets A, B ⊆ R are separated if Ā ∩ B and A ∩ B̄ are both empty. A set E ⊆ R is disconnected if it can be written as E = A ∪ B, where A and B are nonempty separated sets.

A set that is not disconnected is called a connected set.

Connected set (sequential def)

A set E ⊆ R is connected if and only if, for all nonempty disjoint sets A and B satisfying E = A ∪ B, there always exists a convergent sequence (x_n) → x with (x_n) contained in one of A or B, and x an element of the other.

Equivalent notion of connectedness

A set E ⊆ R is connected if and only if whenever a < c < b with a, b ∈ E, it follows that c ∈ E as well.

Functional limit

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

Functional limit (topological version)

Let c be a limit point of the domain of f : A → R. We say lim_x→c f(x) = L provided that, for every ε-neighborhood V_ε(L) of L, there exists a δ-neighborhood V_δ(c) around c with the property that for all x ∈ V_δ(c) different from c (with x ∈ A) it follows that f(x) ∈ V_ε(L).

Sequential criterion for functional limits

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

(i) lim_x→c f(x) = L.

(ii) For all sequences (x_n) ⊆ A satisfying x_n ≠ c and (x_n) → c, it follows that f(x_n) → L.

Algebraic Limit Theorem for functional limits

Let f and g be functions defined on a domain A ⊆ R, and assume lim_x→c f(x) = L and lim_x→c g(x) = M for some limit point c of A. Then,

(i) lim_x→c kf(x) = kL for all k ∈ R,

(ii) lim_x→c [f(x) + g(x)] = L + M,

(iii) lim_x→c [f(x)g(x)] = LM, and

(iv) lim_x→c f(x)/g(x) = L/M, provided M ≠ 0.

Divergence Criterion for functional limits

Let f be a function defined on A, and let c be a limit point of A. If there exist two sequences (x_n) and (y_n) in A with x_n ≠ c and y_n ≠ c and

limx_n = limy_n = c but limf(x_n) ≠ limf(y_n),

then we can conclude that the functional limit lim_x→c f(x) does not exist.

Continuity

A function f : A → R is continuous at a point c ∈ A if, for all ε > 0, there exists a δ > 0 such that whenever |x − c| < δ (and x ∈ A) it follows that |f(x) − f(c)| < ε.

If f is continuous at every point in the domain A, then we say that f is continuous on A.

Characterizations of continuity

Let f : A → R, and let c ∈ A. The function f is continuous at c if and only if any one of the following three conditions is met:

(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) For all V_ε(f(c)), there exists a V_δ(c) with the property that x ∈ V_δ(c) (and x ∈ A) implies f(x) ∈ V_ε(f(c));

(iii) For all (x_n) → c (with x_n ∈ A), it follows that f(x_n) → f(c).

If c is a limit point of A, then the above conditions are equivalent to

(iv) lim_x→c f(x) = f(c).

Criterion for discontinuity

Let f : A → R, and let c ∈ A be a limit point of A. If there exists a sequence (x_n) ⊆ A where (x_n) → c but such that f(x_n) does not converge to f(c), we may conclude that f is not continuous at c.

Algebraic Continuity Theorem

Assume f : A → R and g : A → R are continuous at a point c ∈ A. Then,

(i) kf(x) is continuous at c for all k ∈ R;

(ii) f(x) + g(x) is continuous at c;

(iii) f(x)g(x) is continuous at c; and

(iv) f(x)/g(x) is continuous at c, provided the quotient is defined.

Composition of continuous functions

Given f : A→R and g : B → R, assume that the range f(A) = {f(x) : x ∈ A} is contained in the domain B so that the composition g ◦ f(x) = g(f(x)) is defined on A.

If f is continuous at c ∈ A, and if g is continuous at f(c) ∈ B, then g ◦ f is continuous at c.

Preservation of compact sets

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

Extreme Value Theorem

If f : K → R is continuous on a compact set K ⊆ R, then f attains a maximum and minimum value. In other words, there exist x₀, x₁ ∈ K such that f(x₀) ≤ f(x) ≤ f(x₁) for all x ∈ K.

Uniform continuity

A function f : A → R is uniformly

continuous on A if for every ε > 0 there exists a δ > 0 such that for all x, y ∈ A, |x − y| < δ implies |f(x) − f(y)| < ε.

Sequential criterion for absence of uniform continuity

A function f : A → R fails to be uniformly continuous on A if and only if there exists a particular ε₀ > 0 and two sequences (x_n) and (y_n) in A satisfying

|x_n − y_n| → 0 but |f(x_n) − f(y_n)| ≥ 0.

Uniform continuity on compact sets

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

Continuous Extension Theorem

(i) If f : A → R is uniformly continuous and (x_n) ⊆ A is a Cauchy sequence, then f(x_n) is a Cauchy sequence.

(ii) Let g be a continuous function on the open interval (a, b). g is uniformly continuous on (a, b) if and only if it is possible to define values g(a) and g(b) at the endpoints so that the extended function g is continuous on [a, b].

Intermediate Value Theorem

Let f : [a, b] → 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 ∈ (a, b) where f(c) = L.

Preservation of connected sets

Let f : G → R be continuous. If E ⊆ G is connected, then f(E) is connected as well.

Intermediate Value Property

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

Another way to summarize the Intermediate Value Theorem is to say that every continuous function on [a, b] has the intermediate value property.

Lipschitz functions

A function f : A → R is called Lipschitz if there exists a bound M > 0 such that

|(f(x) − f(y))/(x − y)| ≤ M.

for all x ≠ y ∈ A. Geometrically speaking, a function f is Lipschitz if there is a uniform bound on the magnitude of the slopes of lines drawn through any two points on the graph of f.

If f : A → R is Lipschitz, then it is uniformly continuous on A.

Algebraic Uniform Continuity

Assume that f and g are uniformly continuous functions defined on a common domain A.

Then f(x) + g(x) and f(g(x)) are necessarily uniformly continuous on A as well.

Differentiability

Let g : A → R be a function defined on an interval A. Given c ∈ A, the derivative of g at c is defined by

g′(c) = lim_x→c g(x) − g(c)/(x − c),

provided this limit exists. In this case we say g is differentiable at c. If g′ exists for all points c ∈ A, we say that g is differentiable on A.

Differentiability and continuity

If g : A → R is differentiable at a point c ∈ A, then g is

continuous at c as well.

NOTE: The converse is not necessarily true (i.e. continuity does not guarantee differentiability).

Algebraic Differentiability Theorem

Let f and g be functions defined on an interval A, and assume both are differentiable at some point c ∈ A. Then,

(i) (f + g)′(c) = f ′(c) + g′(c),

(ii) (kf)′(c) = kf ′(c), for all k ∈ R,

(iii) (fg)′(c) = f ′(c)g(c) + f(c)g′(c), and

(iv) (f/g)′ (c) = g(c)f ′(c)−f(c)g′(c)/[g(c)]² , provided that g(c) ∕= 0.

Chain Rule

Let f : A → R and g : B → R satisfy f(A) ⊆ B so that the composition g ◦ f is defined. If f is differentiable at c ∈ A and if g is differentiable at f(c) ∈ B, then g ◦ f is differentiable at c with

(g ◦ f)′(c) = g′(f(c)) · f ′(c).

Interior Extremum Theorem

Let f be differentiable on an open interval (a, b). If f attains a maximum value at some point c ∈ (a, b) (i.e., f(c) ≥ f(x) for all x ∈ (a, b)), then f ′(c) = 0. The same is true if f(c) is a minimum value.

Darboux’s Theorem

If f is differentiable on an interval [a, b], and if α satisfies f ′(a) < α < f ′(b) (or f ′(a) > α > f ′(b)), then there exists a point c ∈ (a, b) where f ′(c) = α.

Rolle’s Theorem

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 ∈ (a, b) where f ′(c) = 0.

Mean Value Theorem

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

f ′(c) = f(b) − f(a)/(b − a).

Differentiability for constant functions

If g : A → R is differentiable on an interval A and satisfies g′(x) = 0 for all x ∈ A, then g(x) = k for some constant k ∈ R.

Functions with the same derivative

If f and g are differentiable functions on an interval A and satisfy

f ′(x) = g′(x) for all x ∈ A, then f(x) = g(x) + k for some constant k ∈ R.

Generalized Mean Value Theorem

If f and g are continuous on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists a point c ∈ (a, b) where

[f(b) − f(a)]g′(c) = [g(b) − g(a)]f ′(c).

If g′ is never zero on (a, b), then the conclusion can be stated as

f ′(c)/g′(c) = (f(b) − f(a))/(g(b) − g(a)).

L’Hospital’s Rule: 0/0 case

Let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of the point a. If f(a) = g(a) = 0 and g′(x) ∕= 0 for all x ∕= a, then

lim _x→a f ′(x)/g′(x) = L implies lim _x→a f(x)/g(x) = L.

Functions with a limit of infinity

Given g : A → R and a limit point c of A, we say that

lim _x→c g(x) = ∞ if, for every M > 0, there exists a δ > 0 such that whenever 0 < |x − c| < δ it follows that g(x) ≥ M.

Similar for the −∞ case.

L’Hospital’s Rule: ∞/∞ case

Assume f and g are differentiable on (a, b) and that g′(x) ∕= 0 for all x ∈ (a, b). If lim _x→a g(x) = ∞ (or −∞), then lim _x→a f ′(x)/g′(x) = L implies lim _x→a f(x)/g(x) = L.

Pointwise Convergence

Let (f_n) be a sequence of functions defined on a set A ⊆ R. Then, (f_n) converges pointwise on A to a limit f defined on A if, for every ε > 0 and x ∈ A, there exists an N ∈ N (perhaps dependent on x) such that |f_n(x) − f(x)| < ε whenever n ≥ N.