Courses / Mathematics / Real Analysis I: completeness, limits and rigorous calculus Mathematics · unit version 1 Real Analysis I: completeness, limits and rigorous calculus Twenty proof-centered lessons developing the real-number completeness property through sequences, topology of the real line, continuity, differentiation and Darboux integration. Written proofs require mathematical review. Function sequences, uniform convergence, metric-space theory and rigorous multivariable analysis belong to the subsequent Analysis II sequence.
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Readiness and preparation Check your starting point Try these short questions on paper, then compare your reasoning. They are optional self-checks, not a placement score or a navigation requirement. Use a targeted review when useful and return with your browser’s Back control.
Negate: for every ε>0 there exists N such that every n≥N satisfies |aₙ−L|<ε. Counterexamples and convergence proofs depend on exact quantifier order.
Compare reasoning and find a refresher There exists ε>0 such that for every N there exists n≥N with |aₙ−L|≥ε. One fixed positive tolerance must defeat every proposed cutoff.
Review negation and witnesses
For ε>0, choose N ensuring 3/n<ε whenever n≥N. A limit proof must turn a requested accuracy into a valid cutoff.
Compare reasoning and find a refresher Any integer N>3/ε works, for example floor(3/ε)+1. The strict inequality matters when 3/ε is an integer.
Review sequence limits
How would you prove A=B for two sets? Topology and completeness arguments frequently require both inclusions.
Compare reasoning and find a refresher Take an arbitrary element of A and prove it lies in B, then take an arbitrary element of B and prove it lies in A. Examples alone do not establish either universal inclusion.
Review double inclusion
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Real Analysis I: construct proofs and test hypotheses
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Analysis manuscript: small area error, different limits
Sustained investigation with separate transfer evidence, saved revisions and human review. It does not change lesson participation requirements.
Lesson sequence Each linked lesson opens on its own page. Expand its details here when you want to check the objectives before choosing.
Lesson details and objectives Separate algebraic order laws from the least-upper-bound property.
Separate algebraic order laws from the least-upper-bound property. Prove uniqueness of least upper bounds from order axioms. Exhibit a rational square-root gap and explain why field laws do not imply completeness. 6 practice tasks.
Lesson details and objectives Use approximation from below and completeness to prove unboundedness of natural numbers.
Use approximation from below and completeness to prove unboundedness of natural numbers. Prove the epsilon characterization of a supremum. Derive the Archimedean property and the supremum rule for sum sets. 8 practice tasks.
Lesson details and objectives Construct rational and irrational points between reals and prove a nested-interval result.
Construct rational and irrational points between reals and prove a nested-interval result. Construct rational and irrational witnesses in an arbitrary open interval. Prove nested-interval existence and separate its uniqueness condition. 8 practice tasks.
Lesson details and objectives Choose a tail index after a tolerance and distinguish convergence from selected samples.
Choose a tail index after a tolerance and distinguish convergence from selected samples. Choose an explicit tail index for a rational sequence. Prove limit uniqueness and negate convergence with the correct quantifiers. 6 practice tasks.
Lesson details and objectives Prove arithmetic laws with boundedness and preserve only justified inequality conclusions.
Prove arithmetic laws with boundedness and preserve only justified inequality conclusions. Prove product and reciprocal limit laws with valid denominator bounds. Distinguish preserved weak inequalities from lost strict inequalities. 6 practice tasks.
Lesson details and objectives Use completeness to extract limits without assuming the original sequence converges.
Use completeness to extract limits without assuming the original sequence converges. Prove bounded monotone convergence from the supremum property. Construct a convergent subsequence by nested bisection with increasing indices. 8 practice tasks.
Lesson details and objectives Control pairs of tail terms and locate exactly where completeness is used.
Control pairs of tail terms and locate exactly where completeness is used. Prove both implications between real convergence and the Cauchy condition. Disprove convergence from shrinking successive differences. 6 practice tasks.
Lesson details and objectives Apply the Cauchy criterion to tails and distinguish absolute from conditional behavior.
Apply the Cauchy criterion to tails and distinguish absolute from conditional behavior. Derive the series Cauchy criterion and absolute-convergence implication. Explain why absolute tails justify arbitrary bijective rearrangements. 6 practice tasks.
Lesson details and objectives Use neighborhoods and sequence witnesses instead of endpoint pictures alone.
Use neighborhoods and sequence witnesses instead of endpoint pictures alone. Prove open-set union and finite-intersection rules using radii. Establish both directions of the sequential characterization of closed sets. 6 practice tasks.
Lesson details and objectives Connect finite subcovers, sequential compactness and closed bounded subsets of R.
Connect finite subcovers, sequential compactness and closed bounded subsets of R. Prove compactness of closed intervals by a bad-half bisection argument. Prove closedness and boundedness are necessary for real compactness. 6 practice tasks.
Lesson details and objectives Build epsilon–delta proofs and separate a punctured limit from the function value.
Build epsilon–delta proofs and separate a punctured limit from the function value. Choose a valid epsilon–delta radius for the square at an arbitrary point. Distinguish continuity, punctured limits and isolated domain points. 6 practice tasks.
Lesson details and objectives Convert failed epsilon–delta statements into explicit sequences of witnesses.
Convert failed epsilon–delta statements into explicit sequences of witnesses. Convert failure of continuity into a sequence with fixed output separation. Use density to prove discontinuity of the rational indicator everywhere. 6 practice tasks.
Lesson details and objectives Distinguish one shared input tolerance from point-dependent tolerances.
Distinguish one shared input tolerance from point-dependent tolerances. Construct paired sequences disproving uniform continuity on an unbounded domain. Prove Heine–Cantor using compact subsequence extraction. 6 practice tasks.
Lesson details and objectives Prove existence statements from compactness and completeness without inventing algorithms.
Prove existence statements from compactness and completeness without inventing algorithms. Prove boundedness and attainment for continuous compact-domain images. Prove the interval intermediate value theorem using a supremum. 6 practice tasks.
Lesson details and objectives Derive global comparison results from local derivatives with endpoint hypotheses.
Derive global comparison results from local derivatives with endpoint hypotheses. Derive Rolle and the mean value theorem from attained extrema. Use derivative signs to prove monotonicity with endpoint hypotheses. 6 practice tasks.
Lesson details and objectives Separate continuity of a derivative from its intermediate-value property and justify remainders.
Separate continuity of a derivative from its intermediate-value property and justify remainders. Prove Darboux's theorem without derivative continuity. Derive the linear Taylor remainder through two applications of Rolle. 10 practice tasks.
Lesson details and objectives Define integration through upper and lower sums rather than antiderivatives.
Define integration through upper and lower sums rather than antiderivatives. Prove refinement and common-refinement inequalities for Darboux sums. Establish the small-gap integrability criterion in both directions. 8 practice tasks.
Lesson details and objectives Prove two distinct sufficient conditions and inspect bounded discontinuous examples.
Prove two distinct sufficient conditions and inspect bounded discontinuous examples. Prove continuous-function integrability using uniform oscillation control. Prove monotone integrability through a telescoping mesh bound. 8 practice tasks.
Lesson details and objectives Distinguish the derivative of accumulation from evaluation using a primitive.
Distinguish the derivative of accumulation from evaluation using a primitive. Prove the derivative of accumulation at a continuity point for both signs of the increment. Derive endpoint evaluation from the mean value theorem. 8 practice tasks.
Lesson details and objectives Treat infinite intervals and singular endpoints as separate limits, not formal substitutions.
Treat infinite intervals and singular endpoints as separate limits, not formal substitutions. Classify power integrals at infinity and at a singular finite endpoint. Distinguish independently convergent tails from symmetric principal values. 8 practice tasks.
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