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← Real Analysis
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Practice deliberately

Work through Real Analysis unit by unit, then test yourself under time.

Practice mode gives one hint and a worked explanation after every question. Test mode mixes all 7 units, holds the explanations until you submit, and then reports which units need another pass. The 8 questions printed further down this page are fixed, so you can read them and their worked answers without starting a set.

Real Analysis coverage and review map

Connect every question to the exact skill it rehearses. Work through the units in order, then return to any topic that still needs a hint or a second attempt.

This page follows the Real Analysis syllabus unit by unit. For shorter mixed retrieval outside the course sequence, use calculus practice questions with answers.

01

Real numbers

Build number sense with exact representations, estimation, and inverse-operation checks before applying the skill in multi-step contexts.

Goal: recognise a real numbers problem from its wording, carry out the governing method, and check that the result is reasonable.

02

Sequences

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a sequences problem from its wording, carry out the governing method, and check that the result is reasonable.

03

Series

Identify the generating pattern, distinguish term values from accumulated sums, and validate formulas with base cases and long-run behavior.

Goal: recognise a series problem from its wording, carry out the governing method, and check that the result is reasonable.

04

Continuity

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a continuity problem from its wording, carry out the governing method, and check that the result is reasonable.

05

Differentiation

Relate local change to slope, apply the appropriate derivative rule, and use sign, units, and graph behavior to interpret the result.

Goal: recognise a differentiation problem from its wording, carry out the governing method, and check that the result is reasonable.

06

Riemann integration

Represent accumulated change, choose bounds and technique carefully, and check the result with units, area, or differentiation.

Goal: recognise a riemann integration problem from its wording, carry out the governing method, and check that the result is reasonable.

07

Metric spaces

Define the central objects, connect at least two representations, solve direct and unfamiliar applications, and explain how the result can be checked.

Goal: recognise a metric spaces problem from its wording, carry out the governing method, and check that the result is reasonable.

Real Analysis questions with worked answers

These 8 questions are printed in full on this page and are drawn across all 7 course units. Nothing here is generated on the fly. Cover the options, solve each one on paper, and only then open the worked answer to compare your method with the one shown.

Question 1: What is the supremum of the set of all numbers 1 - 1/n for positive integers n, and is it in the set?

Real Analysis · Real numbers

  1. supremum 1, not in the set
  2. supremum 1, in the set
  3. supremum 0, in the set
  4. the set has no supremum
Show the worked answer
  1. The values are 0, 1/2, 2/3, 3/4, and so on, always below 1.
  2. So 1 is an upper bound.
  3. No smaller number works, because 1 - 1/n gets as close to 1 as you like by taking n large.
  4. 1 is never reached, since 1/n is never 0, so the supremum is 1 and it is not a member.

Answer: supremum 1, not in the set

Question 2: For a(n) = (3n + 1) / (n + 2), the limit is 3. What is the smallest N with |a(n) - 3| < 0.01 for all n greater than or equal to N?

Real Analysis · Sequences

  1. N = 499
  2. N = 500
  3. N = 498
  4. N = 302
Show the worked answer
  1. Simplify the difference: (3n + 1) / (n + 2) - 3 = (3n + 1 - 3n - 6) / (n + 2) = -5 / (n + 2).
  2. So |a(n) - 3| = 5 / (n + 2).
  3. Require 5 / (n + 2) < 0.01, which means n + 2 > 500, so n > 498.
  4. The smallest integer that works is N = 499. Check: 5 / 501 = 0.00998.

Answer: N = 499

Question 3: For which values of p does the series of 1 / n^p converge?

Real Analysis · Series

  1. p greater than 1
  2. p at least 1
  3. p less than 1
  4. all real p
Show the worked answer
  1. The integral test compares the series with the integral of 1 / x^p from 1 to infinity.
  2. That integral is finite exactly when p is greater than 1.
  3. At p = 1 the series is the harmonic series, which diverges.
  4. For p below 1 the terms shrink even more slowly, so it diverges too.

Answer: p greater than 1

Question 4: Is f(x) = x^2 uniformly continuous on [0, 3], and on all of the real line?

Real Analysis · Continuity

  1. Yes on [0, 3], no on the real line
  2. Yes in both cases
  3. No in either case
  4. No on [0, 3], yes on the real line
Show the worked answer
  1. A continuous function on a closed bounded interval is always uniformly continuous, so [0, 3] is fine.
  2. On the whole line the slope 2x grows without bound.
  3. Take the points n and n + 1/n. Their squares differ by 2 + 1/n^2, which stays above 2.
  4. The inputs get arbitrarily close while the outputs do not, so uniform continuity fails on the real line.

Answer: Yes on [0, 3], no on the real line

Question 5: The Mean Value Theorem is applied to f(x) = x^2 on [1, 4]. What value of c does it produce?

Real Analysis · Differentiation

  1. c = 2.5
  2. c = 2
  3. c = 3
  4. c = 5
Show the worked answer
  1. The average rate of change is (f(4) - f(1)) / (4 - 1) = (16 - 1) / 3 = 5.
  2. The theorem promises some c in (1, 4) with f'(c) equal to 5.
  3. f'(x) = 2x, so 2c = 5.
  4. c = 2.5, which does lie between 1 and 4.

Answer: c = 2.5

Question 6: For f(x) = x on [0, 1] with 4 equal subintervals, what is the upper Riemann sum?

Real Analysis · Riemann integration

  1. 0.625
  2. 0.5
  3. 0.375
  4. 0.75
Show the worked answer
  1. Each subinterval has width 1/4, and f is increasing, so the largest value is at the right end.
  2. The right endpoints are 0.25, 0.5, 0.75, and 1.
  3. Those heights add to 2.5.
  4. 0.25 x 2.5 = 0.625. The exact integral is 0.5, so the upper sum is above it as expected.

Answer: 0.625

Question 7: Which subset of the real line is compact?

Real Analysis · Metric spaces

  1. The closed interval [0, 1]
  2. The half-open interval [0, 1)
  3. The set of all integers
  4. The open interval (0, 1)
Show the worked answer
  1. In the real line, the Heine-Borel theorem says compact means closed and bounded.
  2. [0, 1) is bounded but not closed, since 1 is a limit point that is missing.
  3. The integers form a closed set but are not bounded.
  4. (0, 1) is bounded but not closed. Only [0, 1] is both, so only it is compact.

Answer: The closed interval [0, 1]

Question 8: What does the Bolzano-Weierstrass theorem guarantee for a bounded sequence of real numbers?

Real Analysis · Sequences

  1. It has at least one convergent subsequence.
  2. It converges.
  3. It is monotone.
  4. It is a Cauchy sequence.
Show the worked answer
  1. Bounded does not mean convergent. The sequence 1, -1, 1, -1, ... is bounded and never settles.
  2. But that example does contain a subsequence of all 1s, which converges.
  3. Bolzano-Weierstrass says this always happens: infinitely many terms crowd into some bounded region.
  4. So every bounded real sequence has a convergent subsequence, even when the sequence itself has no limit.

Answer: It has at least one convergent subsequence.

Which unit each printed question belongs to

Use this map after marking your work. If two misses share a unit, review that unit before starting a generated set.

The 8 printed Real Analysis questions, the unit each one tests, and its answer.
QuestionUnitWhat it asksAnswer
1Real numbersWhat is the supremum of the set of all numbers 1 - 1/n for positive integers n, and is it in the set?supremum 1, not in the set
2SequencesFor a(n) = (3n + 1) / (n + 2), the limit is 3. What is the smallest N with |a(n) - 3| < 0.01 for all n greater than or equal to N?N = 499
3SeriesFor which values of p does the series of 1 / n^p converge?p greater than 1
4ContinuityIs f(x) = x^2 uniformly continuous on [0, 3], and on all of the real line?Yes on [0, 3], no on the real line
5DifferentiationThe Mean Value Theorem is applied to f(x) = x^2 on [1, 4]. What value of c does it produce?c = 2.5
6Riemann integrationFor f(x) = x on [0, 1] with 4 equal subintervals, what is the upper Riemann sum?0.625
7Metric spacesWhich subset of the real line is compact?The closed interval [0, 1]
8SequencesWhat does the Bolzano-Weierstrass theorem guarantee for a bounded sequence of real numbers?It has at least one convergent subsequence.

How the generated Real Analysis sets are created

The 8 questions above are fixed and checked. The generator at the top of the page is different: it writes fresh questions with AI from the course and unit information shown here, then the application checks each one for a complete prompt, four choices, one keyed answer, and an explanation. Generated questions are original practice—not official or released exam questions—and AI can still make mathematical mistakes. Verify a disputed answer with the stated method, your course materials, or the MathGPT solver, and follow the site's academic-integrity guidance.

Questions about this Real Analysis practice page

What does this page cover?

It covers all 7 Real Analysis units listed above. Choose one unit for focused work, or mixed review to test method selection, and switch to test mode when you want all units mixed under time.

When can I see correct answers and explanations?

The 8 printed questions on this page keep their worked answers behind a toggle, so you can check any one of them straight away. In the generator above, practice mode explains each question as soon as you answer it, while test mode holds every explanation until you submit.

What should I do with a missed question?

Classify the miss as a definition, setup, calculation, interpretation, or timing error. Re-solve it from a blank page, then use the MathGPT tutor for a hint or method check.