Real Analysis

Introduction

Real analysis is the rigorous study of real numbers, sequences, series, continuity, differentiation, and integration. It provides the theoretical foundation for Calculus, replacing intuitive arguments with precise definitions and proofs. The subject emerged in the 19th century through the work of Cauchy, Weierstrass, Dedekind, and others who sought to place calculus on a firm logical foundation.

The Real Numbers

Construction

The real numbers can be constructed rigorously from the rationals through several equivalent methods:

Dedekind Cuts
A Dedekind cut is a partition of ℚ into two non-empty sets A and B where every element of A is less than every element of B, and A has no greatest element. Each cut defines a real number.

Cauchy Sequences
Define ℝ as equivalence classes of Cauchy sequences of rationals, where two sequences are equivalent if their difference converges to zero.

Completeness Axiom
Alternatively, ℝ can be characterised as the unique complete ordered field—an ordered field where every non-empty bounded set has a supremum.

Properties

The real numbers form an ordered field with the following key properties:

• Archimedean property: For any real x > 0 and y, there exists n ∈ ℕ such that nx > y
• Density of rationals: Between any two reals lies a rational (and an irrational)
• Uncountability: ℝ is uncountable, proved by Cantor’s diagonal argument
• Trichotomy: For any a, b ∈ ℝ, exactly one holds: a < b, a = b, or a > b

Supremum and Infimum

For a non-empty set S ⊆ ℝ:

• Supremum (sup S): The least upper bound—the smallest real number that is ≥ every element of S
• Infimum (inf S): The greatest lower bound—the largest real number that is ≤ every element of S

Completeness Property: Every non-empty subset of ℝ that is bounded above has a supremum in ℝ. This property distinguishes ℝ from ℚ and is equivalent to the completeness axiom.

Characterisation: M = sup S if and only if:

1. M is an upper bound for S
2. For every ε > 0, there exists s ∈ S with s > M - ε

Sequences

Convergence

A sequence (aₙ) converges to limit L if:

$\forall \epsilon >0,\exists N\in \mathbb{N}:n>N⟹|a_n-L|<\epsilon$

We write aₙ → L or lim(n→∞) aₙ = L.

Key facts:

• Limits are unique
• Convergent sequences are bounded
• The converse is false: ((-1)ⁿ) is bounded but divergent

Properties

Algebraic Limit Theorems: If aₙ → a and bₙ → b, then:

• aₙ + bₙ → a + b
• aₙ · bₙ → a · b
• aₙ / bₙ → a / b (provided b ≠ 0)

Squeeze Theorem: If aₙ ≤ bₙ ≤ cₙ and aₙ → L and cₙ → L, then bₙ → L.

Monotone Convergence Theorem: Every bounded monotonic sequence converges.

Bolzano-Weierstrass Theorem: Every bounded sequence has a convergent subsequence.

Subsequences

A subsequence of (aₙ) is a sequence (aₙₖ) where n₁ < n₂ < n₃ < ⋯

Limit Points: L is a limit point of (aₙ) if some subsequence converges to L.

Limit Superior and Inferior:

• lim sup aₙ = inf{sup{aₖ : k ≥ n} : n ∈ ℕ}
• lim inf aₙ = sup{inf{aₖ : k ≥ n} : n ∈ ℕ}

A sequence converges if and only if lim sup aₙ = lim inf aₙ.

Cauchy Sequences

A sequence (aₙ) is Cauchy if:

$\forall \epsilon >0,\exists N:m,n>N⟹|a_m-a_n|<\epsilon$

Key results:

• Every convergent sequence is Cauchy
• Every Cauchy sequence is bounded
• Completeness of ℝ: Every Cauchy sequence in ℝ converges

The last property fails in ℚ, making it incomplete.

Series

Convergence

A series ∑aₙ converges if its sequence of partial sums Sₙ = a₁ + a₂ + ⋯ + aₙ converges.

Types of convergence:

• Absolute convergence: ∑|aₙ| converges
• Conditional convergence: ∑aₙ converges but ∑|aₙ| diverges

Absolute convergence implies convergence, but not conversely (e.g., alternating harmonic series).

Convergence Tests

Comparison Test: If 0 ≤ aₙ ≤ bₙ and ∑bₙ converges, then ∑aₙ converges.

Ratio Test: If |aₙ₊₁/aₙ| → L, then:

• L < 1: converges absolutely
• L > 1: diverges
• L = 1: inconclusive

Root Test: If |aₙ|^(1/n) → L, then same conclusions as ratio test.

Integral Test: If f is positive and decreasing with f(n) = aₙ, then ∑aₙ and ∫f have the same convergence behaviour.

Alternating Series Test: If (aₙ) is positive, decreasing, and aₙ → 0, then ∑(-1)ⁿaₙ converges.

Rearrangements

Riemann Rearrangement Theorem: A conditionally convergent series can be rearranged to converge to any real number, or to diverge.

Absolutely convergent series can be rearranged without changing the sum—this is one reason absolute convergence is often preferred.

Continuity

Definition

A function f: D → ℝ is continuous at c if:

$\forall \epsilon >0,\exists \delta >0:|x-c|<\delta ⟹|f(x)-f(c)|<\epsilon$

Sequential Criterion: f is continuous at c if and only if for every sequence xₙ → c, we have f(xₙ) → f(c).

A function is continuous on a set if it is continuous at every point of that set.

Properties of Continuous Functions

Algebraic Properties: Sums, products, quotients (where defined), and compositions of continuous functions are continuous.

Intermediate Value Theorem: If f is continuous on [a, b] and y lies between f(a) and f(b), then f(c) = y for some c ∈ (a, b).

Extreme Value Theorem: A continuous function on a closed bounded interval [a, b] attains its maximum and minimum values.

Uniform Continuity: f is uniformly continuous if δ can be chosen independently of the point:

$\forall \epsilon >0,\exists \delta >0:|x-y|<\delta ⟹|f(x)-f(y)|<\epsilon$

A continuous function on a closed bounded interval is uniformly continuous.

Types of Discontinuities

• Removable: Both one-sided limits exist and are equal, but differ from f(c)
• Jump: Both one-sided limits exist but are unequal
• Essential: At least one one-sided limit fails to exist

Differentiation

Definition

The derivative of f at c is:

$f^{\prime }(c)={\lim }_{h\to 0}\frac{f(c+h)-f(c)}{h}$

provided this limit exists.

Key result: Differentiability implies continuity (but not conversely—consider |x| at 0).

Mean Value Theorems

Rolle’s Theorem: If f is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then f’(c) = 0 for some c ∈ (a, b).

Mean Value Theorem: If f is continuous on [a, b] and differentiable on (a, b), then:

$f^{\prime }(c)=\frac{f(b)-f(a)}{b-a}$

for some c ∈ (a, b).

Cauchy Mean Value Theorem: If f and g satisfy the above conditions and g’(x) ≠ 0, then:

$\frac{f^{\prime }(c)}{g^{\prime }(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}$

for some c ∈ (a, b). This generalises L’Hôpital’s rule.

Taylor’s Theorem

The Taylor polynomial of degree n for f at a is:

$P_n(x)={\sum }_{k=0}^n\frac{f^{(k)}(a)}{k!}(x-a{)}^k$

Taylor’s Theorem: If f is (n+1)-times differentiable, then f(x) = Pₙ(x) + Rₙ(x), where the remainder Rₙ can be expressed as:

• Lagrange form: Rₙ(x) = f⁽ⁿ⁺¹⁾(ξ)(x - a)ⁿ⁺¹/(n+1)! for some ξ between a and x
• Cauchy form: Rₙ(x) = f⁽ⁿ⁺¹⁾(ξ)(x - ξ)ⁿ(x - a)/n! for some ξ between a and x

Integration

Riemann Integration

Let f be bounded on [a, b]. For a partition P = {x₀, x₁, …, xₙ}:

• Upper sum: U(f, P) = ∑ Mᵢ Δxᵢ where Mᵢ = sup f on [xᵢ₋₁, xᵢ]
• Lower sum: L(f, P) = ∑ mᵢ Δxᵢ where mᵢ = inf f on [xᵢ₋₁, xᵢ]

Darboux Integrability: f is integrable if sup{L(f, P)} = inf{U(f, P)}.

Riemann Integrability: f is integrable if for every ε > 0, there exists a partition P with U(f, P) - L(f, P) < ε.

These definitions are equivalent and give the Riemann integral ∫ₐᵇ f.

Properties

• Linearity: ∫(αf + βg) = α∫f + β∫g
• Additivity: ∫ₐᵇ f + ∫ᵇᶜ f = ∫ₐᶜ f
• Monotonicity: If f ≤ g, then ∫f ≤ ∫g

Fundamental Theorem of Calculus:

1. If f is continuous, then F(x) = ∫ₐˣ f(t)dt is differentiable with F’(x) = f(x)
2. If F’ = f and f is integrable, then ∫ₐᵇ f = F(b) - F(a)

Limitations

The Riemann integral cannot handle:

• Highly discontinuous functions (e.g., Dirichlet function)
• Pointwise limits of integrable functions (may not be integrable)

Lebesgue integration addresses these issues by measuring the domain rather than partitioning the range, leading to more powerful convergence theorems.

Metric Spaces

Definitions

A metric space (X, d) is a set X with a distance function d: X × X → ℝ satisfying:

1. d(x, y) ≥ 0 with equality iff x = y
2. d(x, y) = d(y, x) (symmetry)
3. d(x, z) ≤ d(x, y) + d(y, z) (triangle inequality)

Open ball: B(x, r) = {y ∈ X : d(x, y) < r}

Open set: A set is open if every point has an open ball contained in it.

Closed set: Complement of an open set; equivalently, contains all its limit points.

Convergence: xₙ → x if d(xₙ, x) → 0.

Completeness

A metric space is complete if every Cauchy sequence converges.

Examples:

• ℝ with |x - y| is complete
• ℚ with |x - y| is not complete
• C[a, b] with sup norm is complete

Completion: Every metric space can be embedded in a complete metric space.

Compactness

A set K is compact if every open cover has a finite subcover.

Heine-Borel Theorem: In ℝⁿ, a set is compact if and only if it is closed and bounded.

Sequential Compactness: K is sequentially compact if every sequence has a convergent subsequence with limit in K. In metric spaces, compactness and sequential compactness are equivalent.

Glossary

Term	Definition
Bounded	Contained in some interval [a, b]
Supremum	Least upper bound
Infimum	Greatest lower bound
Cauchy sequence	Terms get arbitrarily close to each other
Convergent	Approaches a finite limit
Uniform continuity	δ works for all points simultaneously
Compact	Every open cover has a finite subcover
Complete	Every Cauchy sequence converges

Learning Resources

Books

• Understanding Analysis by Stephen Abbott — accessible introduction, highly recommended first
• Principles of Mathematical Analysis by Walter Rudin — the classic “Baby Rudin”, more challenging
• Calculus by Michael Spivak — rigorous calculus bridging to analysis
• Real Mathematical Analysis by Charles Pugh — modern approach with good exercises

Courses

• MIT 18.100A Real Analysis
• Francis Su’s Real Analysis lectures

See Also

• Calculus — the computational counterpart to real analysis
• Complex Analysis — extends these ideas to complex numbers
• Mathematical Proofs — proof techniques used throughout analysis