A Primer of Real Analysis

A Primer of Real Analysis by Dan Sloughter provides rigorous foundations beneath familiar calculus operations. Eight chapters begin with sets, order, and completeness, then develop convergence, infinity, topology, continuity, differentiation, integration, and important functions. Computational experience helps but cannot replace precise hypotheses and definitions. Rebuild arguments, track quantifier order, and create counterexamples showing why assumptions matter. The guide describes the linked primer’s conceptual progression and supplies practice instructions; proofs and detailed constructions should be read in the source.

Chapter-by-chapter summaries

1. Fundamentals

Sets, relations, induction, and real-number structure establish the proof language. Completeness distinguishes the real line from systems with gaps and supports existence arguments. An upper bound differs from a maximum, and the least upper bound need not belong to the set. Construct examples for each. When invoking completeness, identify nonemptiness and boundedness explicitly. Familiar numerical intuition becomes useful only when converted into statements precise enough to support subsequent arguments.

2. Sequences and series

Convergence requires eventual closeness expressed through quantifiers. The threshold may depend on demanded accuracy but must work for all later indices. A series converges through its partial-sum sequence, not merely because its terms approach zero. Compare examples illustrating that distinction. Treat monotonicity, boundedness, subsequences, and criteria as tools with hypotheses. Numerical evidence can suggest a limit without controlling infinitely many later terms, which is the proof’s essential task.

3. Cardinality

Bijections compare sets, including infinite collections for which finite intuition fails. A countability argument must actually reach every object; an uncountability argument constructs an object missing from any proposed list. Define the listing or diagonal construction carefully. Explain where the new object differs from each entry. The chapter shows that infinity is not one undifferentiated size and that precise mapping properties can settle questions pictures and enumeration experiments cannot resolve.

4. Topology of the real line

Open and closed sets, neighbourhoods, accumulation, and compactness organize the real line structurally. A set can be neither open nor closed. Verify examples from definitions after drawing them. Compactness supports finite-subcover reasoning and important results for continuous functions. Track boundedness and closedness where used. Similar-looking intervals can have different properties, so visual familiarity cannot justify applying a theorem whose hypotheses the domain does not meet.

5. Limits and continuity

Limits formalize nearby behaviour; continuity additionally requires agreement with the actual value. In quantified proofs, choose an input tolerance guaranteeing the demanded output tolerance. Distinguish pointwise requirements from uniform control where relevant. List domain and regularity hypotheses before applying a theorem. Remove one assumption to seek a counterexample. This reveals why compactness or interval structure can be indispensable rather than decorative conditions attached to a memorized result.

6. Derivatives

Differentiation begins with a limiting quotient, from which rules must be justified. Differentiability implies continuity, while continuity alone is insufficient. Mean-value arguments connect local rates with interval behaviour. Reconstruct a proof and mark where every hypothesis enters. At a suspected singularity, examine the defining quotient rather than trusting a formula derived elsewhere. The chapter replaces procedural familiarity with an account of why differentiation works and where it can legitimately be used.

7. Integrals

Riemann integration formalizes accumulation with partitions and upper and lower control. Existence requires an argument; elementary antiderivative availability is a separate question. Work through a partition and explain how refinement improves estimates. Connect evaluation to the applicable fundamental theorem instead of assuming every integrable function has a convenient closed form. The chapter demonstrates how approximation supports rigorous conclusions and how pointwise irregularity must be assessed through its effect on the defining sums.

8. More functions

Logarithmic, exponential, and trigonometric functions are developed through the analytical foundations rather than imported with all properties assumed. Follow definition, existence, inverse relationships, and derivative results in logical order. Compare this with the teaching order familiar from calculus. Identify possible circularity when a property is used to justify the construction from which that property should follow. The final chapter makes the foundational sequence visible and connects abstract results with central examples.

Study instructions

1. Write quantifier order and one example/nonexample beside every definition.
2. Annotate each theorem proof with where its hypotheses are used.
3. Keep counterexamples organized by missing assumptions.
4. Reconstruct delayed proofs without copying wording, then compare their logical dependencies with the source.

Questions for reflection

• How does a supremum differ from a maximum?
• Why can terms tend to zero while a series diverges?
• Which familiar function properties need proof after construction?

Sources and coverage

This original reading guide covers the linked edition’s main chapters, or the principal sections of its two-chapter calculus text. Consult the source for full arguments, cases, worked examples, exercises, illustrations, and references. Read the source edition.