Real Analysis I: construct proofs and test hypotheses
Bring ideas together without being told which method to use. Show the reasoning that makes your result valid.
Suggested session length: 210 minutes, as an unvalidated planning estimate. This collection contains 19 tasks; choose a manageable set and return for further sessions. Completing the whole collection may take substantially longer. This is a unit assessment, not a full course or AP examination.
Choose your method before opening guidance. State domains, hypotheses and quantifier dependencies; a theorem name or numerical match is not a complete proof.
These public mixed tasks sample the published unit. Saved responses are practice records, not a secure exam result or a guarantee of university readiness.
Written arguments require human mathematical review. Suggested time is planning guidance; the interface is untimed. Explain limitations as well as results.
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Classify the improper integral of sin(3/t²)/t^α on (0,1] for every real α. Derive a signed omitted-tail bound for α<3 and apply it at α=2. Explain why simply reusing the threshold two from sin(1/t) would fail.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Put h=x−1 and G(x)=sin(2h)+3exp(−1/h²) for h≠0, with G(1)=0. Determine its Taylor coefficients at x=1 and prove what the Taylor polynomials converge to on every bounded interval about 1. Does the sum equal G? Compute the discrepancy at x=3/2 and explain why all derivatives at the center fail to detect it.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For f(x)=2x+x² on [0,2], use the partition 0,1/2,2. Compute the integral, right-tag and midpoint sums, their signed errors and valid bounds from |f′|≤6. Find a point attaining the integral average. Explain why averaging the two sampled heights with equal weights gives the wrong quadrature rule.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let E=(−2,−1)∪(3,4] and q(x)=7−2(x−1)². Find the exact image q(E), its infimum/supremum and attainment, with a proof using bounds and approaching members. Compare with the filled set F=(−2,4], which has the same domain infimum and supremum.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let y₀=9 and yₙ₊₁=6−9/yₙ. Prove convergence, derive an exact formula and find the first index with |yₙ−3|<1/10. Then replace y₀ by9/4 and identify the first undefined term. Explain why a strict contraction theorem on [3,9] does not supply the first proof.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For continuous h≥0 on [0,1], prove that ∫₀¹ x²h(x)dx=0 forces h=0, even though the weight vanishes at zero. Replace the weight by max(x−1/2,0) and construct a nonzero continuous nonnegative counterexample. Finally compute the weighted and unweighted integrals of hₙ(x)=max(1−nx,0), n≥2, and explain the limitation of small weighted area.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let φ(t)=cos t on [0,3π] and f(u)=1+2u². Prove and compute the signed substitution integral, the integral with |φ′|, the image-once integral and the integral omitting φ′. Explain why zero derivatives at turning points do not invalidate the forward theorem.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
At c=2, define F(t)=3(t−2)+(t−2)²+(t−2)⁴. Extend G(t)=(F(t)−F(2))/(t−2) continuously at2; compute G(2), G′(2) and the symmetric second quotient of F there. Prove F is convex on R. Then replace F by R(t)=3(t−2)+(t−2)|t−2|: compare the same quantities and diagnose why the symmetric quotient is insufficient.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let α=√3+1/7 and b_n be the fractional part of2nα+1/3, n≥1. Prove its values and every tail are dense in[0,1], and construct a strictly increasing subsequence tending to1. Then replace α by7/5 and find the exact finite set of b_n. Explain why a bounded empty search window cannot establish the irrational conclusion or the rational obstruction.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Using completeness, prove there is a positive real s with s²=5, without assuming square roots already exist. Then use the Archimedean property to prove that between any reals a<b there is a rational number. Give explicit inequalities in both arguments.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove aₙ=(4n+1)/(2n+5) tends to 2 by supplying N(ε). Then prove the product limit theorem for convergent real sequences using an explicit bound, not the invalid claim that the product error equals the product of errors.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Set x₁=3 and xₙ₊₁=(xₙ+5/xₙ)/2. Using r>0 with r²=5, prove the sequence converges to r. Then explain how a Cauchy sequence with a convergent subsequence must converge to the same limit, with an epsilon argument.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove that Σₙ≥1(−1)ⁿ/(n²+3) converges absolutely and that every rearrangement has the same sum. Contrast this with the harmonic series by exhibiting arbitrarily late finite blocks whose sums are bounded below by a fixed positive number.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
For K={0}∪{1/n:n≥1}, identify its limit points and prove compactness directly from an arbitrary open cover. Show that K without 0 is not compact by giving an open cover with no finite subcover. Explain what closedness contributes in the real line.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove f(x)=x³ is uniformly continuous on [−2,2], and disprove uniform continuity on R. Use a fixed positive output tolerance and pairs of arbitrarily close inputs for the second claim. Explain how sequential continuity at one point differs from uniform continuity across a domain.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Show h(x)=x³+x−7 has exactly one real zero r and that 1<r<2. Derive |x−r|≤|h(x)| for every real x. Explain which theorem establishes existence, which establishes uniqueness, and why a minimum of |h| on[1,2] exists.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Suppose f is differentiable on a neighborhood of[a,b] and f′(a)<k<f′(b). Prove some c∈(a,b) has f′(c)=k without assuming f′ is continuous. Separately, construct the quadratic Taylor polynomial for log(1+x) at 0 and bound its error at x=1/5 using a third-derivative bound.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Prove every bounded increasing function on[a,b] is Riemann integrable by bounding upper minus lower Darboux sums. Apply the argument to f(x)=1 for 0≤x<3/10 and f(x)=4 for 3/10≤x≤1; find its integral and explain why the jump value does not prevent integrability.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Let F(x)=∫₀ˣ|t−1/4|dt on[0,1]. Find F′ at interior points, including 1/4, and calculate F(1). Separately evaluate ∫₀¹t^(−3/4)dt as an improper integral, and explain why a symmetric cancellation does not make ∫₋₁¹1/t dt an ordinary convergent improper integral.
Structure your argument and preserve revisions
These optional planning fields save locally. Recording your response saves an immutable copy of the argument and these fields. Older records may contain only first/latest responses; missing intermediate revisions are not reconstructed.
Each field allows 2,000 characters. Keep a specific claim, state where it applies, cite the facts you use, and explain what still needs checking. Blank fields are allowed and are not evidence of mastery.
Review and revisit
Use each rubric to identify a valid argument, a missing condition or a step to revise. For independent evaluation, ask a teacher or reviewer to examine the complete reasoning. Self-review and recorded responses are not verified assessment results.
Coverage and targeted lesson returns
Classify the improper integral of sin(3/t²)/t^α on (0,1] for every real α. Derive a signed omitted-tail bound for α<3 and apply it at α=2. Explain why simply reusing the threshold two from sin(1/t) would fail.
Put h=x−1 and G(x)=sin(2h)+3exp(−1/h²) for h≠0, with G(1)=0. Determine its Taylor coefficients at x=1 and prove what the Taylor polynomials converge to on every bounded interval about 1. Does the sum equal G? Compute the discrepancy at x=3/2 and explain why all derivatives at the center fail to detect it.
For f(x)=2x+x² on [0,2], use the partition 0,1/2,2. Compute the integral, right-tag and midpoint sums, their signed errors and valid bounds from |f′|≤6. Find a point attaining the integral average. Explain why averaging the two sampled heights with equal weights gives the wrong quadrature rule.
Let E=(−2,−1)∪(3,4] and q(x)=7−2(x−1)². Find the exact image q(E), its infimum/supremum and attainment, with a proof using bounds and approaching members. Compare with the filled set F=(−2,4], which has the same domain infimum and supremum.
Let y₀=9 and yₙ₊₁=6−9/yₙ. Prove convergence, derive an exact formula and find the first index with |yₙ−3|<1/10. Then replace y₀ by9/4 and identify the first undefined term. Explain why a strict contraction theorem on [3,9] does not supply the first proof.
For continuous h≥0 on [0,1], prove that ∫₀¹ x²h(x)dx=0 forces h=0, even though the weight vanishes at zero. Replace the weight by max(x−1/2,0) and construct a nonzero continuous nonnegative counterexample. Finally compute the weighted and unweighted integrals of hₙ(x)=max(1−nx,0), n≥2, and explain the limitation of small weighted area.
Let φ(t)=cos t on [0,3π] and f(u)=1+2u². Prove and compute the signed substitution integral, the integral with |φ′|, the image-once integral and the integral omitting φ′. Explain why zero derivatives at turning points do not invalidate the forward theorem.
At c=2, define F(t)=3(t−2)+(t−2)²+(t−2)⁴. Extend G(t)=(F(t)−F(2))/(t−2) continuously at2; compute G(2), G′(2) and the symmetric second quotient of F there. Prove F is convex on R. Then replace F by R(t)=3(t−2)+(t−2)|t−2|: compare the same quantities and diagnose why the symmetric quotient is insufficient.
Let α=√3+1/7 and b_n be the fractional part of2nα+1/3, n≥1. Prove its values and every tail are dense in[0,1], and construct a strictly increasing subsequence tending to1. Then replace α by7/5 and find the exact finite set of b_n. Explain why a bounded empty search window cannot establish the irrational conclusion or the rational obstruction.
Using completeness, prove there is a positive real s with s²=5, without assuming square roots already exist. Then use the Archimedean property to prove that between any reals a<b there is a rational number. Give explicit inequalities in both arguments.
Prove aₙ=(4n+1)/(2n+5) tends to 2 by supplying N(ε). Then prove the product limit theorem for convergent real sequences using an explicit bound, not the invalid claim that the product error equals the product of errors.
Set x₁=3 and xₙ₊₁=(xₙ+5/xₙ)/2. Using r>0 with r²=5, prove the sequence converges to r. Then explain how a Cauchy sequence with a convergent subsequence must converge to the same limit, with an epsilon argument.
Prove that Σₙ≥1(−1)ⁿ/(n²+3) converges absolutely and that every rearrangement has the same sum. Contrast this with the harmonic series by exhibiting arbitrarily late finite blocks whose sums are bounded below by a fixed positive number.
For K={0}∪{1/n:n≥1}, identify its limit points and prove compactness directly from an arbitrary open cover. Show that K without 0 is not compact by giving an open cover with no finite subcover. Explain what closedness contributes in the real line.
Prove f(x)=x³ is uniformly continuous on [−2,2], and disprove uniform continuity on R. Use a fixed positive output tolerance and pairs of arbitrarily close inputs for the second claim. Explain how sequential continuity at one point differs from uniform continuity across a domain.
Show h(x)=x³+x−7 has exactly one real zero r and that 1<r<2. Derive |x−r|≤|h(x)| for every real x. Explain which theorem establishes existence, which establishes uniqueness, and why a minimum of |h| on[1,2] exists.
Suppose f is differentiable on a neighborhood of[a,b] and f′(a)<k<f′(b). Prove some c∈(a,b) has f′(c)=k without assuming f′ is continuous. Separately, construct the quadratic Taylor polynomial for log(1+x) at 0 and bound its error at x=1/5 using a third-derivative bound.
Prove every bounded increasing function on[a,b] is Riemann integrable by bounding upper minus lower Darboux sums. Apply the argument to f(x)=1 for 0≤x<3/10 and f(x)=4 for 3/10≤x≤1; find its integral and explain why the jump value does not prevent integrability.
Let F(x)=∫₀ˣ|t−1/4|dt on[0,1]. Find F′ at interior points, including 1/4, and calculate F(1). Separately evaluate ∫₀¹t^(−3/4)dt as an improper integral, and explain why a symmetric cancellation does not make ∫₋₁¹1/t dt an ordinary convergent improper integral.
