Real Analysis – Comprehensive Reference Guide

Real Number System

Completeness Axiom

Every non-empty set $S \subseteq \mathbb{R}$ bounded above has a least upper bound (supremum).

Supremum & Infimum

$\sup S = M$ iff (1) $\forall x \in S$: $x \leq M$, (2) if $y < M$, $\exists x \in S$: $y < x$.

$\inf S = m$ iff (1) $\forall x \in S$: $x \geq m$, (2) if $y > m$, $\exists x \in S$: $x < y$.

Archimedean Property

$\forall x, y \in \mathbb{R}$ with $x > 0$: $\exists n \in \mathbb{N}$ s.t. $nx > y$.

Consequence: $\mathbb{Q}$ is dense in $\mathbb{R}$.

Bounded Sets

$S$ bounded above: $\exists M$, $x \leq M$ for all $x \in S$.

$S$ bounded: bounded both above and below.

$|S| = \sup|x - y|$ for $x, y \in S$ (diameter).

Sequences: Convergence

Limit Definition

$\lim_{n \to \infty} x_n = L$ iff $\forall \epsilon > 0$, $\exists N$ s.t. $n > N \implies |x_n - L| < \epsilon$.

Cauchy Sequences

$(x_n)$ Cauchy: $\forall \epsilon > 0$, $\exists N$ s.t. $m,n > N \implies |x_m - x_n| < \epsilon$.

In $\mathbb{R}$: Cauchy $\iff$ Convergent.

Monotone Convergence

$(x_n)$ monotone increasing $\& $ bounded above $\implies$ converges to $\sup\{x_n\}$.

$(x_n)$ monotone decreasing $\& $ bounded below $\implies$ converges to $\inf\{x_n\}$.

Squeeze Theorem: If $x_n \leq y_n \leq z_n$ and $x_n, z_n \to L$, then $y_n \to L$.

Limsup & Liminf

Definitions

$\limsup_{n \to \infty} x_n = \lim_{n \to \infty} \sup_{k \geq n} x_k$

$\liminf_{n \to \infty} x_n = \lim_{n \to \infty} \inf_{k \geq n} x_k$

Properties

$\liminf x_n \leq \limsup x_n$

$\lim x_n$ exists $\iff$ $\limsup x_n = \liminf x_n$

$\limsup(x_n + y_n) \leq \limsup x_n + \limsup y_n$

For any sequence, $\limsup x_n$ is the largest accumulation point.

Topology of $\mathbb{R}$

Open & Closed Sets

$U$ open: $\forall x \in U$, $\exists \delta > 0$ with $(x-\delta, x+\delta) \subseteq U$.

$F$ closed: $F^c$ is open; equivalently, $F$ contains all its limit points.

Interior & Closure

$\text{int}(S) = \{x : \exists \delta, (x-\delta,x+\delta) \subseteq S\}$

$\overline{S} = S \cup \{\text{limit points of } S\}$

$S$ closed $\iff$ $S = \overline{S}$

Compactness

Heine-Borel: $K \subseteq \mathbb{R}$ compact $\iff$ closed and bounded.

Open cover: $\{U_\alpha\}$ covers $K$ if $K \subseteq \bigcup U_\alpha$.

Subsequences

Definition

$(x_{n_k})$ subsequence of $(x_n)$ if $n_1 < n_2 < n_3 < \cdots$

Bolzano-Weierstrass

Every bounded sequence has a convergent subsequence.

Accumulation Points

$x$ accumulation point of $\{x_n\}$ iff $\forall \epsilon > 0$, $(x-\epsilon, x+\epsilon)$ contains infinitely many $x_n$.

Equivalently: subseq. $(x_{n_k}) \to x$.

Bounded sequence has finite or infinite acc. points; if infinite, $\limsup$ is the maximum.

Series: Convergence

Series Definition

$\sum_{n=1}^\infty a_n$ converges iff partial sums $S_N = \sum_{n=1}^N a_n$ converge.

Divergence Test

If $\lim a_n \neq 0$, series diverges.

Contrapositive: Convergence requires $a_n \to 0$.

Geometric Series

$\sum r^n = \frac{1}{1-r}$ for $|r| < 1$; diverges for $|r| \geq 1$.

p-Series

$\sum \frac{1}{n^p}$ converges iff $p > 1$.

Careful: $a_n \to 0$ is necessary but not sufficient for convergence.

Series: Convergence Tests

Comparison Test

If $0 \leq a_n \leq b_n$ and $\sum b_n$ converges, so does $\sum a_n$.

If $a_n \geq b_n > 0$ and $\sum b_n$ diverges, so does $\sum a_n$.

Limit Comparison

If $\lim \frac{a_n}{b_n} = c > 0$, then $\sum a_n$ and $\sum b_n$ both converge or diverge.

Ratio Test

Let $L = \limsup \frac{|a_{n+1}|}{|a_n|}$. If $L < 1$, converges; if $L > 1$, diverges.

Root Test

Let $L = \limsup \sqrt[n]{|a_n|}$. If $L < 1$, converges; if $L > 1$, diverges.

Root test stronger than ratio test; use when ratio test inconclusive.

Limits & Continuity

Epsilon-Delta Limit

$\lim_{x \to c} f(x) = L$ iff $\forall \epsilon > 0$, $\exists \delta > 0$ s.t. $0 < |x-c| < \delta \implies |f(x)-L| < \epsilon$.

Continuity

$f$ continuous at $c$ iff $\lim_{x \to c} f(x) = f(c)$.

Equivalently: $\forall \epsilon > 0$, $\exists \delta > 0$ s.t. $|x-c|<\delta \implies |f(x)-f(c)|<\epsilon$.

$f$ continuous on $[a,b]$ if continuous at every point in $[a,b]$.

Uniform Continuity

$f$ uniformly continuous on $S$ iff $\forall \epsilon > 0$, $\exists \delta > 0$ (independent of $x,y$) s.t. $|x-y| < \delta \implies |f(x)-f(y)| < \epsilon$.

Absolute & Conditional Convergence

Definitions

$\sum a_n$ absolutely convergent if $\sum |a_n|$ converges.

$\sum a_n$ conditionally convergent if it converges but $\sum |a_n|$ diverges.

Key Results

Absolute convergence $\implies$ convergence.

If $\sum a_n$ absolutely convergent, any rearrangement converges to same sum.

Alternating Series Test

If $a_n > 0$, $a_n$ decreasing, $a_n \to 0$, then $\sum (-1)^n a_n$ converges.

Error: $|S - S_N| \leq a_{N+1}$.

Rearrangement

Riemann Rearrangement Theorem: Conditionally convergent series can be rearranged to any sum or diverge.

Conditional convergence is fragile: rearrangement changes sum!

Compactness & Connectedness

Properties of Compact Sets

Compact set is closed and bounded.

Closed subset of compact set is compact.

Finite union of compact sets is compact.

Continuous Image

$f$ continuous, $K$ compact $\implies$ $f(K)$ compact.

Corollary: Continuous on compact $\implies$ attains min and max.

Uniform Continuity Theorem

$f$ continuous on compact $\implies$ uniformly continuous.

Connectedness

$S$ connected: cannot write as disjoint union of non-empty open sets.

Connected subsets of $\mathbb{R}$: intervals.

Intermediate Value Theorem: $f$ continuous on interval $\implies$ $f$ takes all intermediate values.

Differentiation

Derivative Definition

$f'(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$

Differentiability Implies Continuity

$f$ differentiable at $c$ $\implies$ $f$ continuous at $c$.

Rolle's Theorem

If $f$ continuous on $[a,b]$, differentiable on $(a,b)$, and $f(a)=f(b)$, then $\exists c \in (a,b)$ with $f'(c)=0$.

Mean Value Theorem

If $f$ continuous on $[a,b]$ and differentiable on $(a,b)$, then $\exists c \in (a,b)$:

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

Interpretation: instantaneous rate equals average rate.

MVT fundamental: connects local (derivative) to global (endpoints).

Advanced Differentiation

Cauchy Mean Value Theorem

If $f,g$ continuous on $[a,b]$, differentiable on $(a,b)$, $g'(x) \neq 0$, then $\exists c$:

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

L'Hôpital's Rule

If $\lim f = \lim g = 0$ (or $\pm\infty$) and $\lim \frac{f'}{g'} = L$, then $\lim \frac{f}{g} = L$.

Requires existence of $\lim f'/g'$ (may not exist while $f/g$ limit does).

Extremum Test

$f'(c) = 0$ and $f''(c) < 0$ $\implies$ local max at $c$.

$f'(c) = 0$ and $f''(c) > 0$ $\implies$ local min at $c$.

Critical points: where $f'=0$ or $f'$ undefined.

Taylor Expansion

Taylor's Theorem

If $f$ is $n$-times differentiable on $[a,b]$, then $\exists c \in (a,b)$:

$$f(b) = \sum_{k=0}^{n-1} \frac{f^{(k)}(a)}{k!}(b-a)^k + \frac{f^{(n)}(c)}{n!}(b-a)^n$$

Taylor Series

$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(a)}{n!}(x-a)^n$$

Converges in neighborhood of $a$ if $R_n(x) \to 0$.

Radius of convergence: $R = \frac{1}{\limsup \left|\frac{f^{(n)}(a)}{n!}\right|^{1/n}}$

Common Series

$e^x = \sum \frac{x^n}{n!}$, $R = \infty$

$\sin x = \sum \frac{(-1)^n x^{2n+1}}{(2n+1)!}$, $R = \infty$

$\frac{1}{1-x} = \sum x^n$, $R = 1$

Riemann Integration

Partition & Sums

Partition: $P = \{a = x_0 < x_1 < \cdots < x_n = b\}$

$\Delta x_i = x_i - x_{i-1}$; $m_i = \inf f(x)$ on $[x_{i-1}, x_i]$

$L(P,f) = \sum m_i \Delta x_i$; $U(P,f) = \sum M_i \Delta x_i$

Riemann Integrability

$f$ Riemann integrable on $[a,b]$ iff $\sup L(P,f) = \inf U(P,f)$.

Then $\int_a^b f = \sup L(P,f) = \inf U(P,f)$

Sufficient Conditions

$f$ continuous on $[a,b]$ $\implies$ integrable.

$f$ monotone on $[a,b]$ $\implies$ integrable.

$f$ bounded with finitely many discontinuities $\implies$ integrable.

Every discontinuity has measure zero for Riemann integrability.

Fundamental Theorem of Calculus

Part 1: Differentiation

If $f$ continuous on $[a,b]$, let $F(x) = \int_a^x f(t)\,dt$. Then $F'(x) = f(x)$.

Part 2: Integration

If $f$ continuous on $[a,b]$ and $G' = f$, then:

$$\int_a^b f = G(b) - G(a)$$

Integration by Parts

$$\int_a^b u\,dv = [uv]_a^b - \int_a^b v\,du$$

Substitution Rule

If $g'$ continuous and $f$ continuous on range of $g$:

$$\int_a^b f(g(x))g'(x)\,dx = \int_{g(a)}^{g(b)} f(u)\,du$$

Integration: Properties

Linearity

$\int (f+g) = \int f + \int g$; $\int cf = c\int f$

Monotonicity

$f \leq g$ on $[a,b]$ $\implies$ $\int_a^b f \leq \int_a^b g$

Additivity

$\int_a^b f + \int_b^c f = \int_a^c f$

Mean Value for Integrals

If $f$ continuous on $[a,b]$, $\exists c \in [a,b]$:

$$\int_a^b f = f(c)(b-a)$$

Bound

$|\int_a^b f| \leq \int_a^b |f| \leq M(b-a)$ where $M = \sup|f|$

Sequences of Functions

Pointwise Convergence

$f_n \to f$ pointwise on $S$ iff $\forall x \in S$: $f_n(x) \to f(x)$.

Uniform Convergence

$f_n \to f$ uniformly on $S$ iff $\forall \epsilon > 0$, $\exists N$ s.t. $n > N \implies |f_n(x) - f(x)| < \epsilon$ for all $x \in S$.

Uniform $\implies$ pointwise, but converse false.

Interchange of Limits

If $f_n \to f$ uniformly and each $f_n$ continuous, then $f$ continuous.

If $f_n \to f$ uniformly on $[a,b]$, then $\int_a^b f_n \to \int_a^b f$.

Pointwise limit of continuous functions need not be continuous!

Function Spaces & Norms

Supremum Norm

$\|f\|_\infty = \sup_{x \in [a,b]} |f(x)|$

$L^p$ Norms

$\|f\|_p = \left(\int_a^b |f|^p\right)^{1/p}$ for $p \geq 1$

Key Spaces

$C([a,b])$: continuous functions on $[a,b]$

$C^k([a,b])$: $k$-times continuously differentiable

$L^p([a,b])$: Lebesgue integrable with $\int |f|^p < \infty$

Completeness

$(C([a,b]), \|\cdot\|_\infty)$ complete (Banach space)

$(L^p([a,b]), \|\cdot\|_p)$ complete for $1 \leq p \leq \infty$

Complete normed space: every Cauchy sequence converges.

Measure Theory Basics

Lebesgue Measure

For $E \subseteq \mathbb{R}$: $m(E) = \inf \sum |I_k|$ over all open covers by intervals.

Properties

$m(\emptyset) = 0$; $m([a,b]) = b-a$

Countable additivity: $m(\bigcup E_k) = \sum m(E_k)$ if disjoint

Monotonicity: $E \subseteq F$ $\implies$ $m(E) \leq m(F)$

Measurable Sets

$E$ measurable if $\forall A$: $m(A) = m(A \cap E) + m(A \setminus E)$

Borel sets measurable; countable intersections/unions of measurable sets measurable

Measure Zero

$E$ has measure zero iff $\forall \epsilon > 0$, $\exists$ open cover with $\sum |I_k| < \epsilon$

Countable sets have measure zero; all derivations define f.a.e.

Not all subsets of $\mathbb{R}$ are measurable (requires Axiom of Choice).

REAL ANALYSIS Comprehensive Reference