Alphabeta Math

Real Analysis

109 pages in 12 parts

A proof-based real analysis collection, built from a construction of the real numbers rather than from axioms taken on trust. Cauchy sequences of rationals and Dedekind cuts each produce a complete ordered field, and from the least upper bound property follow suprema and infima, roots and rational powers, countability, convergent sequences, upper and lower limits, series and the tests that settle them, and the open, closed, compact and connected subsets of the line. Limits and continuity come next, with the intermediate and extreme value theorems and uniform continuity, then the derivative and the mean value theorems, Darboux, L'Hopital, Taylor and convexity, then the Riemann integral, its criterion, the fundamental theorem of calculus, improper integrals, bounded variation and Riemann-Stieltjes. Uniform convergence repairs what pointwise convergence loses, and carries Arzela-Ascoli, Stone-Weierstrass and power series. The exponential, the logarithm, the sine and the cosine are defined by their series, so pi is a theorem rather than a picture. Calculus in several variables closes the collection, with the total derivative, the inverse and implicit function theorems, Jordan content, Fubini, change of variables, arc length and line integrals.

The rest of the library leans on it heavily. Measure theory takes over where the Riemann integral stops, functional analysis starts from the normed spaces and completeness proved here, complex analysis reuses the power series and the rectifiable path, metric topology generalises the sequence and supremum arguments, and differential geometry consumes the total derivative, the inverse and implicit function theorems, Jordan content and the change of variables formula.

Pathway

The parts run in order. Everything a page needs from this group has been read by the time you reach it, and the level on each row is how many dependency steps into the group that page sits.

  1. Part 1 · Building the real numbers

    3 pages

    Analysis needs a field where every bounded set has a least upper bound, and the rationals are not one. Two constructions supply it, one by Cauchy sequences of rationals and one by Dedekind cuts, and both arrive at a totally ordered field with the least upper bound property. The axioms are then stated on their own, so the rest of the subject can quote them without carrying either construction.

    • Construction of the Real Numbers via Cauchy Sequences40 results

      This page constructs the real number field ℝ via Cauchy sequences, starting from the fundamental definitions of the integers and the rational numbers, and proves that ℝ is a totally ordered, complete field.

      11 definitions, 18 lemmas, 8 theorems, 1 example, 2 false statements
    • Construction of the Real Numbers via Dedekind Cuts39 results

      This page constructs the real number field ℝ via Dedekind cuts and proves that ℝ is a totally ordered field with the least-upper-bound property.

      11 definitions, 18 lemmas, 7 theorems, 1 example, 2 false statements
    • Foundations of the Real Numbers for Analysis43 results

      This page assembles the working foundation a first course in real analysis silently assumes: the arithmetic and order facts of ℝ that are usually waved through as "obvious"…

      8 definitions, 18 lemmas, 4 propositions, 4 theorems, 5 corollaries, 1 example, 2 counterexamples, 1 false statement

  2. Part 2 · Order, suprema and countability

    3 pages · after Part 1

    The supremum is how completeness gets used, so it is worth the epsilon characterisations and the behaviour under inclusion, translation, scaling and sums. Roots and rational powers are the first things completeness produces that the rationals cannot. Countability separates the sets that can be listed from those that cannot, and it is what makes the line uncountable.

    • Suprema and Infima17 results

      Completeness of ℝ supplies a least upper bound for every nonempty set bounded above.

      3 definitions, 10 lemmas, 1 theorem, 2 false statements, 1 remarkExamples & counterexamples →
    • Countability and Uncountability20 results

      This page separates the infinite sets of analysis into those that can be listed and those that cannot.

      4 definitions, 5 lemmas, 7 theorems, 2 corollaries, 1 false statement, 1 remark
    • Roots, Rational Powers, and Classical Inequalities22 results

      This page builds exponentiation inside ℝ as far as it can honestly be built with the tools the library currently owns, and then proves the classical inequalities that live at that level.

      3 definitions, 9 lemmas, 7 theorems, 2 false statements, 1 remark

  3. Part 3 · Sequences and completeness

    5 pages · after Part 2

    A sequence converges when its terms eventually stay within any prescribed distance of a limit, and the theorems that follow are the working tools of the subject: the algebra of limits, monotone convergence, Bolzano-Weierstrass, and the equivalence of convergence with the Cauchy condition. Upper and lower limits give every bounded sequence a pair of limits, whether or not it converges. The formal Laurent series field is Cauchy complete without having the least upper bound property, which is why the two forms of completeness are proved equivalent rather than assumed to be the same.

  4. Part 4 · Series

    2 pages · after Part 3

    A series is the limit of its partial sums, so every test is a theorem about sequences. Comparison, condensation, ratio, root and integral tests settle the nonnegative case, where convergence is a question about growth. Absolute convergence is then separated from conditional convergence, and the separation is sharp: rearranging a conditionally convergent series changes what it sums to.

  5. Part 5 · Topology of the line

    2 pages · after Parts 3 and 4

    Open, closed, compact and connected sets let the theorems about functions be stated about the sets they act on rather than about intervals. Compactness is proved in the form used later, and connectedness is what makes the intermediate value theorem an argument about sets. The Cantor set, Baire category and measure zero give two ways for a set to be small, and they do not agree.

  6. Part 6 · Limits and continuity

    3 pages · after Part 5

    Continuity is defined for a function on a subset of the line, then made usable by the sequential criterion, the algebra of continuous functions, and the description by preimages of open sets. The theorems continuity exists for follow: the intermediate value theorem, the extreme value theorem, the image of a compact set, and uniform continuity where the domain forces it. Monotone functions close the part, with a full description of the sets on which a function can fail to be continuous.

  7. Part 7 · Differentiation

    3 pages · after Part 6

    The derivative is a limit of difference quotients, and the mean value theorems are what turn local information about it into global statements about the function. Rolle, the mean value theorem and its Cauchy form come first, then Darboux's theorem, L'Hopital's rule and Taylor's theorem with remainder. Convexity is characterised by the three slope inequality, and through it by monotonicity of the derivative and the sign of the second derivative.

  8. Part 8 · The Riemann integral

    5 pages · after Parts 4, 6 and 7

    Upper and lower Darboux sums define Riemann integrability and yield its linearity, additivity, order properties, standard integrable classes, and the fundamental theorem of calculus. Improper integrals extend integration to unbounded functions and intervals, while Riemann-Stieltjes integration uses integrators of bounded variation. Gauge integration replaces a uniform mesh bound by a positive radius depending on the tag; Cousin's lemma guarantees fine tagged partitions. The Henstock-Kurzweil integral contains the Riemann integral, satisfies a Cauchy criterion, subinterval additivity, and the Saks-Henstock estimate, and integrates every derivative to its endpoint increment. Its compact and noncompact forms support integration by parts, substitution, comparison tests, and Hake extension across a finite missing endpoint.

  9. Part 9 · Sequences of functions and power series

    3 pages · after Parts 3, 7 and 8

    Pointwise convergence of functions preserves neither continuity nor the integral, and uniform convergence is the hypothesis that repairs both. It is set up with the supremum metric, then applied in the space of continuous functions on a compact set, where Arzela-Ascoli describes the compact families and Stone-Weierstrass the dense subalgebras. Power series are the case where uniform convergence is available on every compact subset of the disc of convergence, which is what makes term by term differentiation legitimate.

  10. Part 10 · The classical functions

    7 pages · after Part 9

    The exponential, logarithm, sine and cosine are defined by their power series, so every identity is proved rather than read off a picture. The exponential yields its addition law, derivative and range, and pi comes from the zeros of the cosine, not from geometry. The trigonometric identities, the inverse functions and the integral logarithm follow, and the characterisations an informal treatment assumes are proved equivalent. In several variables they split a Cartesian expression into radial and angular parts, and the standard pathologies follow: line limits without a limit, a circle defeating the vector-valued mean value equality, agreeing mixed partials, differentiability with unbounded or discontinuous partials, uniform boundedness without equicontinuity, an exact solid of revolution, and non-injective spherical coordinates.

  11. Part 11 · Several variables

    12 pages · after Parts 7, 8 and 10

    Norms give limits in Rn; total derivatives give the chain rule, mixed partials, Taylor formulas, extrema, and inverse and implicit functions. Jordan integration, Fubini, change of variables, improper multiple integrals, parameter differentiation, and convexity extend calculus to higher dimensions, while Picard-Lindelof and its consequences settle first-order equations. The added normed-space page makes the ambient completeness theory explicit by introducing normed subspaces, finite product norms, Banach spaces, absolutely convergent series, and completion via bounded linear maps. Manifolds, local flows, partitions of unity, regular surfaces, and constant-rank geometry then close the thread by turning coordinate calculations into intrinsic constructions.

  12. Part 12 · Curves and the fundamental theorems

    10 pages · after Parts 7, 8, 9, 10 and 11

    Path length is the supremum of polygonal approximations, so bounded variation becomes geometry on curves; the fundamental theorems of calculus, line integrals and the gradient theorem turn derivatives back into increments. Jordan content gives area and volume. The tangent-cotangent page rewrites first-order calculus intrinsically on manifolds, with vectors as derivations or curve velocities and differentials as functorial chain-rule maps. The characterisations of π, the complex exponential, oscillatory examples, and the Gamma and Beta integrals connect geometry with special functions. The divergence theorem and classical Stokes then globalize divergence and curl.