# Real Analysis
Real analysis is calculus made honest: every limit, derivative, and
integral you once took on faith is re-derived from a single axiom about
the real numbers, with a proof that says exactly when it holds and when
it fails.
_Figure 001 — A sequence settles into an ε-band around its limit — eventually
as close to L as you like.
_
Everything rests on **completeness**. The rationals have
holes; the reals do not, and the one axiom that fills them — every set
bounded above has a _least_ upper bound — is the engine behind
every convergence theorem that follows.
_Figure 002 — The supremum: the least of all upper bounds of a bounded set.
_
From completeness comes the language of **convergence**.
A sequence has a limit when its terms are eventually trapped in every
tolerance you name, and Cauchy's criterion lets you prove one converges
without knowing the limit in advance.
_Figure 003 — Nested closed intervals shrinking to the single point they share.
_
With limits in hand, **continuity** is the promise that a
function's value never jumps away from where its inputs are heading —
and the theorems that a continuous function on a closed interval attains
its bounds and hits every value between them.
_Figure 004 — Continuity as an unbroken graph, against a jump where one-sided
limits disagree.
_
Differentiation and integration follow the same discipline: the
derivative as a limit of slopes, the Riemann integral as a limit of
sums, and the mean-value and fundamental theorems that bind them
together with proof.
The subtle turn is interchanging limits. Pointwise convergence preserves
almost nothing; **uniform** convergence — one _N_ that
works for every point at once — is what lets you swap a limit with a
derivative or an integral.
_Figure 005 — Uniform convergence: a single ε-tube trapping the whole tail.
_
The same ideas generalize to **metric spaces**, where
distance alone defines open sets, compactness, and completeness, and
the theorems of the real line reappear in their natural setting.
The reward is judgment: you stop asking whether a calculation looks
right and start knowing precisely which hypotheses make it true.
---
## Contents
### 1. Foundations and the Real Number System
1. [Sets, Logic, and Functions](/real-analysis/foundations/sets-logic-functions)
2. [Ordered Fields and the Completeness Axiom](/real-analysis/foundations/ordered-fields-completeness)
3. [Absolute Value, Bounded Sets, and Inequalities](/real-analysis/foundations/absolute-value-bounds)
4. [Intervals, Uncountability, and Decimals](/real-analysis/foundations/intervals-uncountability)
### 2. Sequences and Series
1. [Sequences and Their Limits](/real-analysis/sequences-series/sequences-limits)
2. [Limit Laws and Monotone Convergence](/real-analysis/sequences-series/limit-laws-monotone)
3. [Subsequences, Limit Superior, and Bolzano–Weierstrass](/real-analysis/sequences-series/limsup-bolzano-weierstrass)
4. [Cauchy Sequences and the Completeness of the Reals](/real-analysis/sequences-series/cauchy-completeness)
5. [Series and Convergence Tests](/real-analysis/sequences-series/series-convergence)
6. [Absolute Convergence, the Ratio and Root Tests, and Rearrangements](/real-analysis/sequences-series/absolute-conditional-rearrangement)
### 3. Metric Spaces and Topology
1. [Metric Spaces, Norms, and Examples](/real-analysis/metric-spaces/metric-spaces-norms)
2. [Open and Closed Sets, Interior, Closure](/real-analysis/metric-spaces/open-closed-sets)
3. [Convergence, Cauchy Sequences, and Completeness](/real-analysis/metric-spaces/convergence-completeness)
4. [Compactness](/real-analysis/metric-spaces/compactness)
5. [Connectedness](/real-analysis/metric-spaces/connectedness)
### 4. Limits and Continuity
1. [Limits of Functions](/real-analysis/continuity/limits-of-functions)
2. [Continuous Functions](/real-analysis/continuity/continuous-functions)
3. [Extreme and Intermediate Value Theorems](/real-analysis/continuity/evt-ivt)
4. [Uniform Continuity](/real-analysis/continuity/uniform-continuity)
5. [Continuity on Metric Spaces](/real-analysis/continuity/continuity-metric-spaces)
6. [Limits at Infinity and Monotone Functions](/real-analysis/continuity/limits-infinity-monotone)
### 5. Differentiation
1. [The Derivative](/real-analysis/differentiation/the-derivative)
2. [The Mean Value Theorem](/real-analysis/differentiation/mean-value-theorem)
3. [Taylor's Theorem](/real-analysis/differentiation/taylors-theorem)
4. [The Inverse Function Theorem in One Variable](/real-analysis/differentiation/inverse-function-1d)
### 6. The Riemann Integral
1. [Partitions, Darboux Sums, and Integrability](/real-analysis/riemann-integration/darboux-integral)
2. [Which Functions Are Integrable](/real-analysis/riemann-integration/integrability-classes)
3. [Properties of the Integral](/real-analysis/riemann-integration/properties-of-the-integral)
4. [The Fundamental Theorem of Calculus](/real-analysis/riemann-integration/fundamental-theorem)
5. [The Logarithm, Exponential, and Improper Integrals](/real-analysis/riemann-integration/log-exp-improper)
### 7. Sequences and Series of Functions
1. [Pointwise and Uniform Convergence](/real-analysis/function-sequences/pointwise-uniform-convergence)
2. [Interchange of Limits: Continuity, Integration, Differentiation](/real-analysis/function-sequences/interchange-of-limits)
3. [Power Series and the Weierstrass Approximation Theorem](/real-analysis/function-sequences/power-series-weierstrass)
4. [Picard's Existence and Uniqueness Theorem](/real-analysis/function-sequences/picard-ode)
### 8. Functions of Several Variables (Introduction)
1. [The Derivative of a Map ℝⁿ → ℝᵐ](/real-analysis/several-variables/differentiability-rn)
2. [Directional Derivatives, the Gradient, and the Chain Rule](/real-analysis/several-variables/gradient-chain-rule)
3. [Higher Derivatives, Taylor's Theorem, and Extrema](/real-analysis/several-variables/higher-derivatives-taylor-extrema)
4. [The Inverse and Implicit Function Theorems](/real-analysis/several-variables/inverse-implicit-theorems)
5. [Multiple Integrals](/real-analysis/several-variables/multiple-integrals)