Seton Hall University



Real Analysis Highlights You need to be able to state or explain each of the following statementsChapter 1 to 3:De Morgan LawsEuclid’s TheoremEquivalence RelationsThe IntegersThe RationalsCardinalityCountable and uncountableCantor’s diagonalization argumentCardinality of N, Z, Q, R, RxR, algebraic numbers, transcendental numbers, irrational numbersCardinality of countable unions of countable sets, finite cross products of countable sets, countable cross products of countable sets, power set of a setContinuum hypothesisHilbert’s HotelCantor Bernstein TheoremInductionPartially ordered, ordered, well orderedUpper bound, lower bound, least upper bound, largest lower boundLeast upper bound property of RArchimedian PropertySequenceConvergence of a sequenceMonotone or bounded sequenceCauchy sequenceLim sup and lim infMonotone and bounded sequences convergeEuler’s numberCompleteness theorem of RSubsequenceBolzano Weierstrass theoremPower sequenceExponent sequenceRoot n of n sequenceNth root sequenceBinomial sequenceEuler’s sequenceExponential sequenceChapter 4: Series Series, including convergence of a seriesAbsolute and conditional convergenceRearranging absolutely and conditionally convergent seriesFor fun: Leaning Tower of LireFor fun: Zeno’s ParadoxConvergence TestsDivergence Test Comparison Test Limit Comparison TestRoot Test Ratio Test Alternating Series Test Integral TestSpecial SeriesGeometric Series (incl. proof)Harmonic Series (incl. proof)Alternating Harmonic Series p SeriesEuler’s SeriesChapter 5: TopologyOpen and Closed SetsUnions and Intersections of Open and Closed SetsCharacterizing open setsBoundary Point and Interior PointIsolated Point and Accumulation PointClosed sets, accumulation points, and sequencesCompact SetsHeine Borel TheoremIntersection of Nested Compact SetsPerfect SetsPerfect Sets are UnountableConnected and Disconnected setsThe Cantor Setis perfectuncountable but zero lengthdoes not contain any open setis totally disconnectedChapter 6Limit of a function (sequence version)Limit of a function (epsilon-delta version)ContinuityUniform ContinuityContinuity vs Uniform ContinuityContinuity preserves LimitsUniform Continuity preserves Cauchy sequencesTypes of DiscontinuityMonotone functions and discontinuitiesCharacterization of Discontinuity of the Second KindContinuity and Topology (continuous functions and inverse images of sets)Images of compact sets and of connected setsMax/Min Theorem for continuous functionsBolzano TheoremIntermediate Value TheoremDerivativeDerivative as linear approximationDifferentiability and ContinuityProduct, Quotient, and Chain RuleRolle’s Theorem and Mean Value TheoremL’Hospital RulesA function that is not continuous at any point in R (Dirichlet function)A function that is continuous at the irrational numbers and discontinuous at the rational numbers.A function that is differentiable, but the derivative is not continuous A function that is n-times differentiable, but not (n+1)-times differentiable (Cn function)A function that is not zero, infinitely often differentiable, but the n-th derivative at zero is always zero (Cinf function)A function that is continuous everywhere and nowhere differentiable in R (Weierstrass function)Chapter 7:Partition and Riemann SumUpper and Lower SumUpper and Lower Riemann IntegralRiemann IntegralRiemann’s LemmaA function that is not Riemann integrableContinuous functions and the IntegralAlmost Continuous Functions and the IntegralMonotone Functions and the IntegralProperties of the Riemann IntegralFundamental Theorem CalculusIntegral Evaluation ShortcutA function f such that axftdt is not differentiableAntiderivativeSubstitution RuleIntegration by PartsMean Value Theorem for IntegrationPartial Fraction DecompositionBe sure to know how to evaluate the following integrals:sec2xdx and 11+x2 dx and 11-x2 dxtanxdx and lnxdxx2cosxdxexcos?(x) dxsin3xdx11-x4 dx ................
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