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MATH40002 Seven Year Effective Coverage Matrix REVISED

English review edition prepared on 4 October 2026 from a preserved source copy. It is a later presentation, not the historical study interface. Source checks and difficulty judgements describe the original material author's own process; they do not indicate Imperial College London endorsement.

Source SHA-256: e88eb23a9e2da58ecd57ed0a67be1f73eff4e076c82895c3dab37baaa855aeca
Source date: 2026-08-06

EffectiveCoverage

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ability_unit_idyearquestionmarksabilityS1Q1S1Q2S1Q3S1Q4S2Q1S2Q2S2Q3S2Q4S3Q1S3Q2S3Q3S3Q4S4Q1S4Q2S4Q3S4Q4S5Q1S5Q2S5Q3S5Q4S6Q1S6Q2S6Q3S6Q4best_revised_grade
2020-1(a)(i)20201(a)(i)1State the definition of countably infinite preciselyDDDDBDDDBDDDADDDDDDDBDDDA
2020-1(a)(ii)20201(a)(ii)1Determine the cardinality of the set of binary sequencesDDDDBDDDBDDDADDDDDDDBDDDA
2020-1(a)(iii)20201(a)(iii)1Determine the cardinality of eventually-zero sequencesDDDDBDDDBDDDADDDDDDDBDDDA
2020-1(a)(iv)20201(a)(iv)1Count sequences with a fixed tailDDDDDDDDBDDDADDDDDDDDDDDA
2020-1(b)20201(b)4Recognise an infimum from quantified conditionsDDDDDDDBBDDDADDDDDDDBDBDA
2020-1(c)(1)20201(c)(1)1Translate quantified structure into a propertyCDDDCBDDCCDDADDDDDDDCDDDA
2020-1(c)(2)20201(c)(2)1Translate quantified structure into a propertyCDDDCBDDCCDDADDDDDDDCDDDA
2020-1(c)(3)20201(c)(3)1Translate quantified structure into a propertyCDDDCBDDCCDDADDDDDDDCDDDA
2020-1(c)(4)20201(c)(4)1Translate quantified structure into a propertyCDDDCBDDCCDDADDDDDDDCDDDA
2020-1(c)(5)20201(c)(5)1Translate quantified structure into a propertyCDDDCBDDCCDDADDDDDDDCDDDA
2020-1(d)(i)20201(d)(i)1State the definition of convergenceCDDDDBDDCDDDADDDDDDDCDDDA
2020-1(d)(ii)20201(d)(ii)1State the definition of divergenceDDDDDDDDDDDDADDDDDDDDDDDA
2020-1(d)(iii)20201(d)(iii)1Give a divergent sequence with consecutive differences tending to zeroCDDDDDDDDDDDACDDDDDDDCDDA
2020-1(d)(iv)20201(d)(iv)4Control the full sequence using its even subsequence and consecutive differencesCDDDDBDDCDDDACDDDDDDCCDDA
2020-2(a)20202(a)2Prove an upper bound involving radicalsDDDDCDCDDDDDDADDCDDDDDDDA
2020-2(b)20202(b)2Estimate the difference of consecutive square rootsDDDDCDCDDDDDDADDCDDDDDDDA
2020-2(c)(i)20202(c)(i)2Define convergence of a seriesCDDDDDDDDDDDDADDDDDDDCDDA
2020-2(c)(ii)20202(c)(ii)7Prove from first principles sum 1/sqrt(n) divergenceCDDDDDDDDDDDDADDDDDDDCDDA
2020-2(d)20202(d)7From a_n→a<1 prove sum a_n^n ConvergenceCDDDDBDDCDDDDADDDDDDCDDDA
2020-3(a)(i)20203(a)(i)2A positive limit implies eventual positivityCDDDDBDDCDDDDDDDDDDDCADDA
2020-3(a)(ii)20203(a)(ii)3Limit of a finite moving geometric meanDDDDCDCDDDDDDDDDCDDDDADDA
2020-3(a)(iii)20203(a)(iii)2Construct divergence with convergent moving geometric meansCDDDDDDDDDDDDCDDDDDDDADDA
2020-3(b)(i)20203(b)(i)2State the alternating-series testADDDDDDDDDDDDADDDDDDDDDDA
2020-3(b)(ii)20203(b)(ii)5Recognise that the alternating-series test is inapplicable and prove divergenceADDDDDDDDDDDDADDDDDDDDDDA
2020-3(b)(iii)20203(b)(iii)6Find the radius while avoiding a failed direct ratio testADDDDDDDDDCDDADDDDDDDDDDA
2020-4(a)(i)20204(a)(i)3Prove continuity at an oscillation pointDDDDDDDDDDDDDDADDCDDDDDDA
2020-4(a)(ii)20204(a)(ii)2Continuity of a composition at nonzero pointsDDDDDDDDDDDDDDADDBDDDDDDA
2020-4(b)(i)20204(b)(i)2State the intermediate value theoremDDDDDDDDDDDDDDADDDDDDDDDA
2020-4(b)(ii)20204(b)(ii)3Prove that an equation has a rootDDDDDDDDDDDDDDADDDDDDDDDA
2020-4(b)(iii)20204(b)(iii)3Count zeros using alternating signsDDDDDDDDDDDDDDADDDDDDDDDA
2020-4(c)(i)20204(c)(i)4An absolute-value contraction condition implies a zeroDDDDDDDDDDDDDDADDDDDDDDDA
2020-4(c)(ii)20204(c)(ii)3Construct a discontinuous counterexample with no zerosDDDDDDDDDDDDDDADDDDDDDDDA
2020-5(a)(i)20205(a)(i)2State the mean value theoremDDDDDDDDDDDDDDADDDDDDDADA
2020-5(a)(ii)20205(a)(ii)3use MVT prove Bernoulli type inequalityDDBCDDDDDDDDDDADDDBDDDADA
2020-5(a)(iii)20205(a)(iii)4If the function has a finite limit and its derivative has a limit, the latter is 0DDDDDDDCDDDCDDADDDDDDDADA
2020-5(b)(i)20205(b)(i)3Calculate the second-order Taylor PolynomialDDBDDDDDDDDDDDADDDBDDDDBA
2020-5(b)(ii)20205(b)(ii)2Use the second derivative to determine convexityDDBBDDDDDDDDDDADDDBDDDDDA
2020-5(b)(iii)20205(b)(iii)2Derive a triangle inequality from convexityDDBBDDDDDDDDDDADDDBDDDDDA
2020-5(c)20205(c)4Prove using a uniform remainder bound Taylor Uniform convergence of polynomialsDDCDDDDDDDDDDDADDDCDDDDAA
2020-6(a)(i)20206(a)(i)1Determine and prove / counterexampleDDACDDDCDDDCDDDADDDCDDDCA
2020-6(a)(ii)20206(a)(ii)1Determine and prove / counterexampleDDACDDDCDDDCDDDADDDCDDDCA
2020-6(a)(iii)20206(a)(iii)1Determine and prove / counterexampleDDACDDDCDDDCDDDADDDCDDDCA
2020-6(a)(iv)20206(a)(iv)1Determine and prove / counterexampleDDACDDDCDDDCDDDADDDCDDDCA
2020-6(a)(v)20206(a)(v)2Determine and prove / counterexampleDDACDDDCDDDCDDDADDDCDDDCA
2020-6(b)(i)20206(b)(i)3State FTCDDDBDDDDDDDDDDDADDDBDDDBA
2020-6(b)(ii)20206(b)(ii)3Calculate upper and lower sums for a monotone function on an integer partitionDDCDDDDDDDDDDDDADDDDDDDDA
2020-6(b)(iii)20206(b)(iii)3Bound large finite sums using integralsDDDBDDDDDDDDDDDADDDBDDDBA
2020-6(c)20206(c)5Equality on a dense set implies equal integralsDDCDDDDDDDDDDDDADDDDDDDDA
2021-1(a)20211(a)5Decide using set inclusion Venn Empty regionDDDDDCCDDCDDDDDDDDDDBDDDB
2021-1(b)(i)20211(b)(i)1Determine whether a power set can be countably infiniteDDDDBDDDBDDDBDDDDDDDBDDDB
2021-1(b)(ii)20211(b)(ii)1Determine whether a power set can be uncountableDDDDBDDDBDDDBDDDDDDDBDDDB
2021-1(c)(i)20211(c)(i)1Construct a sequence with no convergent rearrangementDDDDDDDDDDDDDDDDDDDDDDDDD
2021-1(c)(ii)20211(c)(ii)1Determine whether value sets are countably infiniteDDDDDDDDDDDDDDDDDDDDDDDDD
2021-1(c)(iii)20211(c)(iii)1Conditional convergence permits rearrangement to any real sumDDDDDDDDDDDDDDDDDDDDDDDDD
2021-1(d)(i)20211(d)(i)1Counterexample with no convergent subsequenceDDDDBBDDDDDDBDDDDDDDDDDDB
2021-1(d)(ii)20211(d)(ii)1Unbounded but possessing a convergent subsequenceDDDDBBDDDDDDBDDDDDDDDDDDB
2021-1(d)(iii)20211(d)(iii)1Counterexample with consecutive differences tending to zeroCDDDDDDDDDDDDCDDDDDDDCDDC
2021-1(d)(iv)20211(d)(iv)1Ratio tends to 1/2 Implies the terms tend to 0CDDDDBDDCDDDDDDDDDDDCDDDB
2021-1(e)20211(e)6Hadamard Lower bound on the product's radiusDDDDDDDDDDCDDDDDDDDDDADDA
2021-2(a)(i)20212(a)(i)3Prove from first principles that a rational expression tends to 0CDDDDBDDCDDDDDDDDDDDCDDDB
2021-2(a)(ii)20212(a)(ii)3Prove from first principles that cubing preserves a limitDDDDCDCDDDDDDDDDCDDDDDDDC
2021-2(b)20212(b)6Construct a divergent sequence with convergent weighted averagesCDDDCDBDDDDDDCDDCDDDDCDDB
2021-2(c)20212(c)5Prove the supremum formula for a quotient setDDDDCDCBBDDDBDDDCDDDADBDA
2021-2(d)20212(d)3From dyadic Convergence of a subsequence of partial sums implies convergence of the entire seriesDDDDDDDDDDDDDDDDDDDDDDDDD
2021-3(a)20213(a)6Prove convergence of a radical-sum sequenceDDDDCDCDDCDDDDDDCDDDBDDDB
2021-3(b)20213(b)5Extract from an unbounded sequence reciprocals tending to 0 SubsequenceDDDDBBDDDDDDADDDDDDDDDDDA
2021-3(c)(i)20213(c)(i)4prove sup(a_n+b_n) inequalityDDDDDDDBBDDDBDDDDDDDBDBDB
2021-3(c)(ii)20213(c)(ii)2Give a counterexample with strict inequalityCDDDDDDBBDDDBCDDDDDDBCBDB
2021-3(d)20213(d)3The series of prefix averages of a convergent positive-term sequence divergesCDDDCDBDDDDDDCDDCDDDDADDA
2021-4(a)(i)20214(a)(i)3Oscillating function at 0 ContinuousDDDDDDDDDDDDDDADDCDDDDDDA
2021-4(a)(ii)20214(a)(ii)3in 0 DifferentiateDDDDDDDDDDDDDDADDDDDDDDDA
2021-4(b)(i)20214(b)(i)3Parameterised equation has a rootDDDDDDDDDDDDDDADDDDDDDDDA
2021-4(b)(ii)20214(b)(ii)2Construct polynomials with no intersectionDDDDDDDDDDDDDDADDDDDDDDDA
2021-4(c)(i)(1)20214(c)(i)(1)1Classify sets as open or closedDDDDDDDBDDDDDDADDADDDDDDA
2021-4(c)(i)(2)20214(c)(i)(2)1Classify sets as open or closedDDDDDDDBDDDDDDADDADDDDDDA
2021-4(c)(i)(3)20214(c)(i)(3)1Classify sets as open or closedDDDDDDDBDDDDDDADDADDDDDDA
2021-4(c)(i)(4)20214(c)(i)(4)1Classify sets as open or closedDDDDDDDBDDDDDDADDADDDDDDA
2021-4(c)(i)(5)20214(c)(i)(5)1Classify sets as open or closedDDDDDDDBDDDDDDADDADDDDDDA
2021-4(c)(ii)20214(c)(ii)4f' >0 Image preserves opennessDDCDDDDDDDDDDDADDADDDDDDA
2021-5(a)(i)20215(a)(i)2Construct a bounded continuous function with no local extremaDDDDDDDCDDDCDDDDDDADDDCDA
2021-5(a)(ii)20215(a)(ii)4n+1 Equal-value points imply n Zeros of the derivative of orderDDDDDDDDDDDDDDDDDDADDDDDA
2021-5(b)(i)20215(b)(i)2Constant derivative implies linearityDDDDDDDDDDDDDDDDDDADDDDDA
2021-5(b)(ii)20215(b)(ii)3Two intersections imply the derivative equals the secant slopeDDDDDDDDDDDDDDDDDDADDDDDA
2021-5(b)(iii)20215(b)(iii)3Upper and lower bounds for arctangentDDDDDDDDDDDDDDDDDDADDDDDA
2021-5(c)(i)20215(c)(i)2find 12 Coefficients of derivatives of orderDDCDDDDDDDDDDDCDDDADDDDCA
2021-5(c)(ii)20215(c)(ii)4Prove the midpoint inequalityDDBBDDDDDDDDDDBDDDADDDDDA
2021-6(a)(i)20216(a)(i)2Write upper and lower sums for an equally spaced partitionDDCDDDDDDDDDDDDCDDDDDDDDC
2021-6(a)(ii)20216(a)(ii)2Find the difference between upper and lower sumsDDDDDDDDDDDDDDDDDDDDDDDDD
2021-6(a)(iii)20216(a)(iii)2From Darboux Bound power sumsDDDBDDDDDDDDDDDDDDDBDDDBB
2021-6(b)(i)20216(b)(i)3compare e^{-x^3}DCDDDDDDDDDDDDDDDDDDDDDAA
2021-6(b)(ii)20216(b)(ii)3f(x)/x→1 Derive a limit of integral averages 1/2DDDCDDDDDDDDDDDDDDDCDDDAA
2021-6(c)(i)20216(c)(i)4Find local extremaDDDBDDDBDDDBDDDDDDDCDDBCB
2021-6(c)(ii)20216(c)(ii)4Prove that the limit at infinity does not existDDDCDDDDDDDDDDDDDDDCDDDCC
2022-1(a)(i)20221(a)(i)3False: unboundedness does not imply that the reciprocal tends to 0CDDDDDDDDDDDDCDDDDDDDCDDC
2022-1(a)(ii)20221(a)(ii)3Supremum of a discrete bounded set belongs to the setDDDDCCDBBDDDBDDDDDDDBDBDB
2022-1(a)(iii)20221(a)(iii)3A series selected by an injective indexing map remains absolutely convergentDDDDDDDDDDDDDDDDDDDDDDDDD
2022-1(b)(i)20221(b)(i)3The set of bivariate diagonally periodic functions is uncountableDDDDBDDDBDDDBDDDDDDDADDDA
2022-1(b)(ii)20221(b)(ii)3The set of functions periodic only in the first variable is uncountableDDDDBDDDBDDDBDDDDDDDADDDA
2022-1(c)(i)20221(c)(i)3Find alternating discrete sets supDDDDDDDBBDDDAADDDDDDBABDA
2022-1(c)(ii)20221(c)(ii)2A convergent series multiplied by 1/n^2 becomes absolutely convergentDCDDDDDDDDCDAADDDDDDDADDA
2022-2(a)(i)20222(a)(i)2Bounded but not Cauchy ExampleCDDDDBDDDBDDDCDDDDDDDCDDB
2022-2(a)(ii)20222(a)(ii)2Nonmonotone with no convergent subsequenceCDDDBBDDDDDDBCDDDDDDDCDDB
2022-2(a)(iii)20222(a)(iii)2All terms irrational and every real number a subsequential limitDDDDBCDDBDDDBDDDDDDDBDDDB
2022-2(b)(i)20222(b)(i)2Positive-term tail sums decrease to 0DDDDDDDDDDDDDDDDADDDDDDDA
2022-2(b)(ii)20222(b)(ii)3sum a_n/sqrt(r_n) ConvergenceDCDDDDDDDDCDDDDDADDDDDDDA
2022-2(b)(iii)20222(b)(iii)6sum a_n/r_n divergenceCDDDDDDDDDDDDCDDADDDDCDDA
2022-2(b)(iv)20222(b)(iv)3sum r_n<∞ infer n a_n→0DCDDDDDDDDCDDDDDADDDDDDDA
2022-3(a)(i)20223(a)(i)3sqrt(6+a_n) ConvergenceDDDDDDCDDCDDDDDDDDDDBDDDB
2022-3(a)(ii)20223(a)(ii)4Geometric decay of consecutive differences implies convergenceDDDDDDDDDDDDDDDDCDDDDDDDC
2022-3(b)(i)20223(b)(i)3Prove from first principles that a fraction tends to 1CDDDDBDDCDDDDDDDDDDDCDDDB
2022-3(b)(ii)20223(b)(ii)4max(a_n,b_n) LimitDDDDCDCDDDDDDDDDCDDDDDDDC
2022-3(c)20223(c)6a_{n+1}<=a_n+1/n^2 Implies convergenceDDDDCCCDDCDDBDDDDDDDBDDDB
2022-4(a)(i)20224(a)(i)2cos(1/x) Continuous at nonzero pointsDADDDDDDDDDDDDDDDBDDDDDDA
2022-4(a)(ii)20224(a)(ii)30 Cannot extend continuously atDADDDDDDDDDDDDDDDDDDDDDDA
2022-4(b)(i)20224(b)(i)2x=cos^3x Has a solutionDDDDDDDDDDDDDDDDDDDDDDDDD
2022-4(b)(ii)20224(b)(ii)3Alternating integer values imply infinitely many zerosDDDDDDDDDDDDDDDDDDDDDDDDD
2022-4(c)(i)20224(c)(i)3A strictly increasing discontinuous function with a noncompact image of a compact setDDCDDDDBDDDDDDDDDADDDDDDA
2022-4(c)(ii)20224(c)(ii)3A strictly increasing continuous function maps open sets to open setsDDCDDDDDDDDDDDDDDADDDDDDA
2022-4(c)(iii)20224(c)(iii)4f(x)/x boundedDDCDDDDDDDDDDDDDDADDDDDDA
2022-5(a)(i)20225(a)(i)2Stationary point that is not an extremumDDDDDDDCDDDCDDADDDADDDCDA
2022-5(a)(ii)20225(a)(ii)2f'→0 but f→∞DDDDDDDDDDDDDDADDDADDDDDA
2022-5(a)(iii)20225(a)(iii)2f Bounded but f' unboundedDDDDDDDDDDDDDDADDDADDDDDA
2022-5(b)(i)20225(b)(i)3sqrt(cos x) Second-order polynomialDDBDDDDDDDDDDDADDDBDDDDBA
2022-5(b)(ii)20225(b)(ii)2Midpoint inequalityDDBBDDDDDDDDDDADDDBDDDDDA
2022-5(b)(iii)20225(b)(iii)3First order Taylor Give a numerical lower boundDDBDDDDDDDDDDDADDDBDDDDBA
2022-5(c)(i)20225(c)(i)2Bounds on logarithmic secant slopesDDDDDDDDDDDDDDADDDADDDDAA
2022-5(c)(ii)20225(c)(ii)4Four collinear points imply a zero of the third derivativeDDDDDDDDDDDDDDADDDADDDDAA
2022-6(a)(i)20226(a)(i)4log Upper and lower sums on an exponential partitionDDCDDDDDDDDDDDDADDDDDDDDA
2022-6(a)(ii)20226(a)(ii)2find U-L Closed formDDDDDDDDDDDDDDDADDDDDDDDA
2022-6(a)(iii)20226(a)(iii)4U Tends to the integral 1DDDBDDDDDDDDDDDADDDBDDDBA
2022-6(b)(i)20226(b)(i)3e^{1/x}-1 divergenceDCDDDDDDDDDDDDDADDDDDDDDA
2022-6(b)(ii)20226(b)(ii)3r>1 Converges whenDCDDDDDDDDDDDDDADDDDDDDDA
2022-6(c)20226(c)4All moments equal 0 deduce f=0DDCDDDDDDDCDDDCADDCDDDDCA
2023-1(a)(i)20231(a)(i)5Decreasing N to N The set of functions is countableDDDDBDDDADDDBDDDDDDDBDDDA
2023-1(a)(ii)20231(a)(ii)5The set of increasing functions is uncountableDDDDBDDDADDDBDDDDDDDBDDDA
2023-1(b)(i)20231(b)(i)4Rational expression tends to 0CDDDDBDDADDDDDDDDDDDCDDDA
2023-1(b)(ii)20231(b)(ii)6Equivalent characterisation by definitionDDDDDDDBADDDBDDDDDDDBDBDA
2023-2(a)(i)20232(a)(i)3Diverges to +∞ definitionCDDDDDDDDADDDCDDADDDDCDDA
2023-2(a)(ii)20232(a)(ii)5If the positive part diverges and the negative part converges, every rearrangement tends to +∞DDDDDDDDDADDDDDDADDDDDDDA
2023-2(b)20232(b)5An approximate monotonicity inequality implies convergenceDDDDDDCDDADDDDDDCDDDBDDDA
2023-2(c)20232(c)7A nonmonotone recurrence converges; find its limitDDDDDDDDDADDDDDDCDDDDDDDA
2023-3(a)20233(a)5Convergence of adjacent geometric means does not imply convergence of the original seriesDDDDDDDDDDDDDDDDDDDDDDDDD
2023-3(b)20233(b)5Convergence of the original positive-term series implies convergence of the series of adjacent geometric meansDDDDDDDDDDDDDDDDDDDDDDDDD
2023-3(c)20233(c)5A bounded sequence whose convergent subsequences share a limit convergesDDDDBBDDDDDDBDDDDDDDDDDDB
2023-3(d)20233(d)5Fixed-step differences tend to 0 Does not imply convergenceCDDDDDDDDDDDDCDDDDDDDCDDC
2023-4(a)(i)20234(a)(i)2epsilon-delta definitionDDDDDDDDDDADDDCDDDDDDDCDA
2023-4(a)(ii)20234(a)(ii)5Necessary and sufficient sequential criterionDCDDDDDDDDADDDCDDCDDDDCDA
2023-4(a)(iii)20234(a)(iii)3Rational / A function defined on irrational points at a≠0 No limitDCDDDDDDDDADDDDDDCDDDDDDA
2023-4(b)(i)20234(b)(i)2cos(1/x)sin(1/x) No limitDDDDDDDDDDADDDDDDDDDDDDDA
2023-4(b)(ii)20234(b)(ii)2x Multiplied by an oscillation tends to 0DDDDDDDDDDADDDDDDCDDDDDDA
2023-4(b)(iii)20234(b)(iii)3sin(ax)/x→aDDDDCDCDDDADDDCDCDDDDDCDA
2023-4(c)20234(c)3Limits everywhere for a function with finitely layered exceptional points 0DBDDDDDDDDADDDDDDCDDDDDDA
2023-5(a)20235(a)3State the definition of discontinuityDDDDDDDDDDCDBDCDDDDDDDCDB
2023-5(b)(i)20235(b)(i)3Continuous, strictly monotone and unbounded above and below implies bijectiveDDDDDDDCDDDCDDDDDDDDDDCDC
2023-5(b)(ii)20235(b)(ii)4A continuous function with finite limits at both ends is boundedDDDDDDDCDDDCDDDDDDDDDDCDC
2023-5(b)(iii)20235(b)(iii)4Equal limits at both ends imply at least one global extremumDDDDDDDCDDDCDDDDDDDDDDCDC
2023-5(b)(iv)20235(b)(iv)3A continuous monotone function bounded below need not attain its minimumDDDDDDDBDDDBDDDDDDDDDDBDB
2023-5(c)20235(c)3[0,1] A continuous self-map has a fixed pointDDDDDDDDDDADDDDDDDDDDDADA
2023-6(a)20236(a)4ODE Identically zero between two zerosDDDCDDDDDDDDDDDDDDDADDDDA
2023-6(b)20236(b)5Changing one value preserves integrability and the integral 0DDCCDDDCDDDADDDCDDDADDDCA
2023-6(c)20236(c)6Integral against every test function equals 0 deduce f=0DDDDDDDDDDDDDDDCDDDADDDDA
2023-6(d)20236(d)5Differentiate both sides with the same initial valueDDDBDDDDDDDDDDDDDDDADDDBA
2024-1(a)20241(a)5a_n→0 and sum b_n Convergence does not imply sum a_nb_nCDDDDDDDDDDDDCDDDDDDDCDDC
2024-1(b)20241(b)5A set of functions with nonempty finite range cannot be countably infiniteDDDDBDDDBDDDBDDDDDDDBDDDB
2024-1(c)20241(c)5Consecutive differences tend to 0 Does not imply CauchyCDDDDDDDDDDDDCDDDDDDDCDDC
2024-1(d)20241(d)5Absolute convergence implies sum(a_n)^n ConvergenceDDDDDDDDDDDDDDDDDDDDDDDDD
2024-2(a)(i)20242(a)(i)3Definition of convergenceCDDDDADDADDDDDDDDDDDCDDDA
2024-2(a)(ii)20242(a)(ii)6Reciprocal limit from first principlesCDDDCACDADDDDDDDCDDDCDDDA
2024-2(b)20242(b)5sup(AB)=supA supBDDDDCDCBADDDBDDDCDDDBDBDA
2024-2(c)20242(c)6|a_n| Does not tend to ∞ Then there is a convergent subsequenceDDDDCADDACDDBDDDDDDDDDDDA
2024-3(a)(i)20243(a)(i)6a_n→0 Implies the averages →0CDDDCDADDADDDCDDCDDDDCDDA
2024-3(a)(ii)20243(a)(ii)2Original sequence diverges while its averages convergeCDDDCDADDADDDCDDCDDDDCDDA
2024-3(a)(iii)20243(a)(iii)6n Differences tend to 0 Implies the difference between averages and original terms tends to 0CCDDCDADDACDDCDDCDDDDCDDA
2024-3(b)20243(b)6a_{n+1}=b+a_n^2 Necessary and sufficient for convergenceDDDDDDDDDADDDDDDDDDDDDDDA
2024-4(a)(i)20244(a)(i)2epsilon-deltaDDDDDDDDDDADDDCDDDDDDDCDA
2024-4(a)(ii)20244(a)(ii)3Negation of quantifiersDDDDDDDDDDADBDCDDDDDDDCDA
2024-4(a)(iii)20244(a)(iii)3Negation of quantifiersDDDDDDDDDDADBDCDDDDDDDCDA
2024-4(a)(iv)20244(a)(iv)3epsilon-delta Two pointsDDDDDDDDDDADDDDDDDDDDDDDA
2024-4(b)(i)20244(b)(i)2Continuity implies sequential continuityDADDDDDDDDADDDDDDCDDDDDDA
2024-4(b)(ii)20244(b)(ii)3Discontinuous everywhereDADDDDDDDDADDDDDDCDDDDDDA
2024-4(b)(iii)20244(b)(iii)4Sequential continuity implies continuityDADDDDDDDDADDDCDDCDDDDCDA
2024-5(a)(i)20245(a)(i)2Continuity need not imply global boundednessDDDDDDDBDDDBDDDDDADDDDBDA
2024-5(a)(ii)20245(a)(ii)3Boundedness need not imply uniform continuityDDDDDDDDDDDDDDDDDADDDDDDA
2024-5(a)(iii)20245(a)(iii)3Uniform continuity need not imply boundednessDDDDDDDDDDDDDDDDDADDDDDDA
2024-5(a)(iv)20245(a)(iv)2Bounded derivative implies LipschitzDDDDDDDDDDDDDDDDDADDDDDDA
2024-5(b)(i)20245(b)(i)4Finite limits at both ends + Continuity implies boundednessDDDDDDDCDDDADDDDDDDDDDCDA
2024-5(b)(ii)20245(b)(ii)3Finite limits at both ends and no critical pointsDDDDDDDBDDDADDDDDDDDDDBDA
2024-5(b)(iii)20245(b)(iii)3Equal limits at both ends imply f'=0DDDDDDDCDDDADDDDDDDDDDBDA
2024-6(a)20246(a)4Changing one point preserves integrabilityDDCCDDDCDDDADDDCDDDCDDDCA
2024-6(b)(i)20246(b)(i)4Equally spaced upper and lower sums of a monotone functionDDCADDDDDDDADDDCDDDDDDDDA
2024-6(b)(ii)20246(b)(ii)4Darboux Definition of integrabilityDDCADDDDDDDADDDCDDDDDDDDA
2024-6(b)(iii)20246(b)(iii)3U-L Endpoint difference /nDDDADDDDDDDADDDDDDDDDDDDA
2024-6(b)(iv)20246(b)(iv)5Every monotone function is integrable on a closed intervalDDDADDDBDDDADDDBDDDBDDDBA
2025-1(a)20251(a)6Tends to 0 The set of rational sequences is uncountableDDDDADDDBDDDBDDDDDDDBDDDA
2025-1(b)20251(b)4Every subseries convergesDDDDADDDDDDDDDDDDDDDDDDDA
2025-1(c)20251(c)3Infinitely many terms lie in [-1,1] Then there is a convergent subsequenceDDDDABDDDDDDBDDDDDDDDDDDA
2025-1(d)20251(d)7Terms tend to 0 and partial sums repeatedly cross [0,1] Then every r Has a subsequenceCDDDADCDDBDDDCDDDDDDDCDDA
2025-2(a)(i)20252(a)(i)3Definition of boundednessDDDDDADDDCDDDDDDDDDDDDDDA
2025-2(a)(ii)20252(a)(ii)3Cauchy definitionDDDDDADDDCDDDDDDDDDDDDDDA
2025-2(a)(iii)20252(a)(iii)4Cauchy Implies boundednessDDDDDADDDCDDDDDDDDDDDDDDA
2025-2(b)20252(b)5a_n→a,b_n→0 Implies the product →0CDDDCACDCDDDDDDDCDDDCDDDA
2025-2(c)20252(c)5Bounded and all convergent subsequences share a Then the full sequence tends to aDDDDBADDDDDDBDDDDDDDDDDDA
2025-3(a)(i)20253(a)(i)5sum a_n Convergent and sum a_n^2 Divergence implies conditional convergenceDDDDDDADDDDDDDDDDDDDDDDDA
2025-3(a)(ii)20253(a)(ii)51/n sum j a_j→0CCDDCDADDDCDDCDDCDDDDCDDA
2025-3(b)20253(b)5A convergent sequence under N bijective permutation retains its limitCDDDDDADDDDDDDDDDDDDDDDDA
2025-3(c)20253(c)5Harmonic block sums convergeDDDDDDADDCDDDDDDDDDDBDDDA
2025-4(a)(i)20254(a)(i)2Unions of open sets are openDDDDDDDADDDDDDDDDCDDDDDDA
2025-4(a)(ii)20254(a)(ii)2Zeros at 0 Infinitely many nearby zeros do not imply identically zero in a neighbourhoodDDDDDDDADDDDDDDDDCDDDDDDA
2025-4(a)(iii)20254(a)(iii)2A finite limit implies boundedness in a neighbourhoodDDDDDDDADDCDDDCDDCDDDDCDA
2025-4(a)(iv)20254(a)(iv)2Discontinuous everywhere g Constant composition f Can be continuousDDDDDDDADDDDDDDDDCDDDDDDA
2025-4(b)20254(b)2Quantified definition of nonexistence of a limitDDDDDDDADDCDBDCDDDDDDDCDA
2025-4(c)20254(c)5Rational function at 4 Limit 1DDDDDDDADDCDDDCDDDDDDDCDA
2025-4(d)20254(d)5A monotone function's left limit equals the left-hand value set's supDDCDDDDABDDDBDDDDDDDBDADA
2025-5(a)(i)20255(a)(i)2f''>0 Does not ensure an interior minimumDDABDDDADDDCDDBDDDBDDDCDA
2025-5(a)(ii)20255(a)(ii)2The intermediate value property does not imply continuityDDADDDDADDDDDDDDDDDDDDDDA
2025-5(a)(iii)20255(a)(iii)2A convex function on a closed interval may be discontinuous at endpointsDDABDDDADDDDDDBDDDBDDDDDA
2025-5(a)(iv)20255(a)(iv)2Interior differentiability and an endpoint difference alone do not ensure a positive derivativeDDADDDDADDDDDDDDDDDDDDDDA
2025-5(b)20255(b)5f''(x) Precise definition of existenceDDDDDDDADDDDDDDDDDDDDDDDA
2025-5(c)20255(c)3Derivative of an even function at 0 is 0DDCDDDDADDDDDDCDDDCDDDADA
2025-5(d)20255(d)4Equal finite limits at both ends imply a critical pointDDDDDDDADDDADDDDDDDDDDBDA
2025-6(a)(i)20256(a)(i)2Every partition L=U Implies a constantDDCDDDDADDDDDDDCDDDDDDDDA
2025-6(a)(ii)20256(a)(ii)2Nonnegative integrable function with zero integral need not be constantDDDBDDDADDDBDDDBDDDBDDDBA
2025-6(a)(iii)20256(a)(iii)2R A concave function is locally continuous and hence integrable on a closed intervalDDBBDDDADDDCDDBCDDBCDDDCA
2025-6(a)(iv)20256(a)(iv)2Refinement cannot decrease the lower sumDDCDDDDADDDDDDDCDDDDDDDDA
2025-6(b)20256(b)4Changing one point preserves integrabilityDDDBDDDADDDBDDDBDDDBDDDBA
2025-6(c)20256(c)5Prove a piecewise function integrable and evaluate its integralDDDBDDDADDDBDDDBDDDBDDDBA
2025-6(d)20256(d)3A nonnegative continuous function has zero integral exactly when it is identically zeroDDDDDDDADDDDDDDCDDDDDDDAA
2026-1(a)(i)20261(a)(i)4a_n=sqrt(n+1)-sqrt(n) Tends to 0ADDDDBDDCDDDDDDDDDDDCDDDA
2026-1(a)(ii)20261(a)(ii)2sqrt(n)a_n→1/2ADDDDBDDCDDDDDDDDDDDCDDDA
2026-1(b)20261(b)3sum a_n divergenceADDDDDDDDDDDDCDDDDDDDCDDA
2026-1(c)20261(c)5Convergence and remainder boundADDDDDDDDDDDDDDDDDDDDDDDA
2026-1(d)(i)20261(d)(i)3Absolute convergence inside the circle , z=1 divergence , R=1ADDDDDDDDDCDDDDDDDDDDDDDA
2026-1(d)(ii)20261(d)(ii)3On the unit circle except 1 conditionally convergent elsewhereADDDDDDDDDDDDDDDDDDDDDDDA
2026-2(a)20262(a)6Values on dense rational points equal 0 Implies identically 0DADDDDDDDDDDDDDDDCDDDDDDA
2026-2(b)(i)20262(b)(i)4Finite one-sided limits give a continuous extension to the closed intervalDADDDDDDDDDDDDDDDDDDDDDDA
2026-2(b)(ii)20262(b)(ii)4From Heine-Cantor Implies uniform continuity on the open intervalDADDDDDDDDDDDDDDDDDDDDDDA
2026-2(c)(i)20262(c)(i)2Oscillating function is continuous and boundedDADDDDDDDDDDDDDDDCDDDDDDA
2026-2(c)(ii)20262(c)(ii)4Construct two sequences approaching the same endpoint whose function values differ by a fixed amountDADDDDDDDDDDDDDDDDDDDDDDA
2026-3(a)(i)20263(a)(i)3Compositions of integrable functions need not be integrableDDACDDDCDDDCDDDCDDDCDDDCA
2026-3(a)(ii)20263(a)(ii)3max of convex convexDDACDDDDDDDDDDCDDDCDDDDDA
2026-3(a)(iii)20263(a)(iii)3A bijection differentiable in both directions cannot have derivative 0DDADDDDDDDDDDDCDDDCDDDCDA
2026-3(a)(iv)20263(a)(iv)3Rational points x^2 Otherwise 0 in 0 Not twice differentiableDDADDDDDDDDDDDDDDDDDDDDDA
2026-3(b)(i)20263(b)(i)2Darboux Integrability and upper and lower sumsDDADDDDDDDDDDDDCDDDDDDDDA
2026-3(b)(ii)20263(b)(ii)3Taylor Integral remainderDDADDDDDDDDDDDBDDDBDDDDBA
2026-3(b)(iii)20263(b)(iii)3L'Hopital RuleDDADDDDDDDDDDDDDDDDDDDDDA
2026-4(a)20264(a)5A bounded function with finitely many discontinuities is integrableDDCADDDCDDDCDDDCDDDCDDDAA
2026-4(b)(i)20264(b)(i)2f NonincreasingDDBADDDDDDDDDDBDDDBCDDDDA
2026-4(b)(ii)20264(b)(ii)3use inf Use the definition to prove existence of a limitDDDADDCBBCDCBDDDDDDDBDBDA
2026-4(b)(iii)20264(b)(iii)4f'→0DDDADDDDDDDDDDDDDDDCDDDDA
2026-4(b)(iv)20264(b)(iv)4use f' Integrate monotonicity to obtain two-sided boundsDDBADDDDDDDDDDBDDDBCDDDCA
2026-4(b)(v)20264(b)(v)2x f'(x)→0DDDADDDCDDDCDDDDDDDCDDCDA