Chapter 1
Preliminaries
Triangle inequality: |x + y| ≤ |x| + |y|
Reverse triangle inequality: ||x| − |y|| ≤ |x − y|
Axiom of completeness
Every nonempty set of real numbers that is bounded above has a least upper bound. s is a least upper
bound for A:
1. s is an upper bound for A (a ≤ b for all a ∈ A)
2. if b is an upper bound for A, then s ≤ b
Lemma: s − ϵ < a for all ϵ > 0 and a ∈ A.
Consequences of completeness
If In is a sequence of closed intervals, then their intersection is nonempty
Archimedian property: there exists an n ∈ N and an x ∈ R such that n < x and n1 < x
Given any two real numbers a and b where a < b, there exists an irrational number t such that a < t < b
and a rational number r such that a < r < b.
Cardinality
Injective (one-to-one): f (a) = f (b) =⇒ a = b
Surjective (onto): f (a) = b for all a ∈ A and b ∈ B.
Bijective if both injective and surjective.
A has the same cardinality as B if there exists a bijective function f : A → B. Write as A ∼ B
A set A is countable if S ∼ A for S ⊆ N
The union of countable sets must be countable.
Cantors theorem
Power set: P (A) refers to the collection of all subsets of A
Given any set A, there does not exist a function f : A → P (A) that is onto.
A countable ⇐⇒ f : A → N injective
A countable ⇐⇒ g : N → A surjective
If B is countable and f : A → B is injective, then A is countable
If A is countable and g : A → B is surjective, then B is countable.
Chapter 2
Limit of a sequence
A sequence (an ) converges to a, if, for all ϵ > 0, there exists an N ∈ N such that whenever n ≥ N , it follows
that |an − a| < ϵ
Vϵ (a) = {x ∈ R : |x − a| < ϵ}
The limit of a sequence, when it exists, must be unique.
1
, The algebraic and order limit theorems
A sequence is bounded if there exists an M > 0 such that |xn | < M for all n ∈ N. Every convergent series
is bounded.
Let lim an = a and lim bn = b. Then
an ≥ 0 for all n ∈ N =⇒ a ≥ 0
an ≤ bn for all n ∈ N =⇒ a ≤ b
If there exists a c ∈ R for which c ≤ an for all n ∈ N =⇒ c ≤ a
Some standard limits:
lim n1a = 0 for (a > 0) lim cn = 0 for |c| < 1
n a
√
lim c n = 0 for |c| < 1 and a ∈ R lim n c = 1 for c > 0
√
lim m n = 1 lim nn!m = 0
Monotone convergence theorem
A sequence is increasing if an ≤ an+1 and decreasing if an ≥ an+1 . It is monotone if it is either increasing
or decreasing.
If a sequence is monotone and bounded, then it converges.
We say that an infinite series converges to b if its sequence of partial sums
P∞ converges to b P∞ n
Cauchy condensation
P∞ 1 test: Suppose (b n ) decreases and bn ≥ 0. Then, n=1 bn converges ⇐⇒ n=0 2 b2n
The series n=1 np converges if and only if p > 1
P∞ that f : [1, ∞) → RR∞is positive, continuous and monotonically decreasing. Let
Integral test: Assume
ak = f (x). Then, k=1 ak converges ⇐⇒ 1 f (x)dx < ∞
Bolzano-weierstrass theorem
A subsequence of (an ) is denoted ((an )k ).
Subsequences of a convergent sequence converge to the same limit as the original sequence.
Every bounded sequence contains a convergent subsequence.
If different subsequences have different limits, then (an ) diverges.
The cauchy criterion
A sequence (an ) is called a cauchy sequence if, for every ϵ > 0, there exists an N ∈ N such that whenever
m, n ≥ N it follows that |an − am | < ϵ
A sequence converges if and only if it is a cauchy sequence.
2
Preliminaries
Triangle inequality: |x + y| ≤ |x| + |y|
Reverse triangle inequality: ||x| − |y|| ≤ |x − y|
Axiom of completeness
Every nonempty set of real numbers that is bounded above has a least upper bound. s is a least upper
bound for A:
1. s is an upper bound for A (a ≤ b for all a ∈ A)
2. if b is an upper bound for A, then s ≤ b
Lemma: s − ϵ < a for all ϵ > 0 and a ∈ A.
Consequences of completeness
If In is a sequence of closed intervals, then their intersection is nonempty
Archimedian property: there exists an n ∈ N and an x ∈ R such that n < x and n1 < x
Given any two real numbers a and b where a < b, there exists an irrational number t such that a < t < b
and a rational number r such that a < r < b.
Cardinality
Injective (one-to-one): f (a) = f (b) =⇒ a = b
Surjective (onto): f (a) = b for all a ∈ A and b ∈ B.
Bijective if both injective and surjective.
A has the same cardinality as B if there exists a bijective function f : A → B. Write as A ∼ B
A set A is countable if S ∼ A for S ⊆ N
The union of countable sets must be countable.
Cantors theorem
Power set: P (A) refers to the collection of all subsets of A
Given any set A, there does not exist a function f : A → P (A) that is onto.
A countable ⇐⇒ f : A → N injective
A countable ⇐⇒ g : N → A surjective
If B is countable and f : A → B is injective, then A is countable
If A is countable and g : A → B is surjective, then B is countable.
Chapter 2
Limit of a sequence
A sequence (an ) converges to a, if, for all ϵ > 0, there exists an N ∈ N such that whenever n ≥ N , it follows
that |an − a| < ϵ
Vϵ (a) = {x ∈ R : |x − a| < ϵ}
The limit of a sequence, when it exists, must be unique.
1
, The algebraic and order limit theorems
A sequence is bounded if there exists an M > 0 such that |xn | < M for all n ∈ N. Every convergent series
is bounded.
Let lim an = a and lim bn = b. Then
an ≥ 0 for all n ∈ N =⇒ a ≥ 0
an ≤ bn for all n ∈ N =⇒ a ≤ b
If there exists a c ∈ R for which c ≤ an for all n ∈ N =⇒ c ≤ a
Some standard limits:
lim n1a = 0 for (a > 0) lim cn = 0 for |c| < 1
n a
√
lim c n = 0 for |c| < 1 and a ∈ R lim n c = 1 for c > 0
√
lim m n = 1 lim nn!m = 0
Monotone convergence theorem
A sequence is increasing if an ≤ an+1 and decreasing if an ≥ an+1 . It is monotone if it is either increasing
or decreasing.
If a sequence is monotone and bounded, then it converges.
We say that an infinite series converges to b if its sequence of partial sums
P∞ converges to b P∞ n
Cauchy condensation
P∞ 1 test: Suppose (b n ) decreases and bn ≥ 0. Then, n=1 bn converges ⇐⇒ n=0 2 b2n
The series n=1 np converges if and only if p > 1
P∞ that f : [1, ∞) → RR∞is positive, continuous and monotonically decreasing. Let
Integral test: Assume
ak = f (x). Then, k=1 ak converges ⇐⇒ 1 f (x)dx < ∞
Bolzano-weierstrass theorem
A subsequence of (an ) is denoted ((an )k ).
Subsequences of a convergent sequence converge to the same limit as the original sequence.
Every bounded sequence contains a convergent subsequence.
If different subsequences have different limits, then (an ) diverges.
The cauchy criterion
A sequence (an ) is called a cauchy sequence if, for every ϵ > 0, there exists an N ∈ N such that whenever
m, n ≥ N it follows that |an − am | < ϵ
A sequence converges if and only if it is a cauchy sequence.
2
