MH3100: pass the exams, not just read the notes
Your complete guide to Nanyang Technological University's real analysis i course. See where the marks are, work real practice questions, and study with an AI tutor that knows MH3100.
Sia generates MH3100 practice questions, walks through the real numbers and sequences: convergence step by step, and quizzes you on the material the exam weights most heavily.
Worked example
A sequence of continuous functions converges pointwise to a limit function. Why is that not enough to conclude the limit is continuous, and what does uniform convergence add?
Write both definitions and note where the quantifiers sit. Pointwise convergence fixes a point first and then asks for an index beyond which the values are close. Uniform convergence asks for a single index that works for every point at once. The order of the quantifiers is the entire distinction.
Build the failure. The standard counterexample is a sequence of continuous functions on a closed interval converging pointwise to a function with a jump. Each function in the sequence is continuous; the limit is not. Pointwise convergence is satisfied and continuity is lost.
See what uniform convergence buys. With a single index valid across the domain, the usual three-part estimate closes: approximate the limit by a term of the sequence uniformly, use the continuity of that term locally, and return. This is why NTU places sequences of functions in week 11, after continuity has been developed properly in weeks 8 and 9.
The trap: Option B is the error the week exists to prevent, and it is not carelessness — it is the natural assumption if you met these ideas in a calculus course where the distinction was never drawn. This is exactly what the published course aim means by understanding better many results from Calculus I and II which you studied without proofs: the theorems you used were true under hypotheses you were never shown, and week 11 is where one of those hypotheses becomes visible. classic slip!
One exam decides 50% of your grade. Summative assessment. This whole page is built around that.
Overview
What MH3100 is, and where it sits
MH3100 is the first of NTU's two real analysis courses and it is compulsory for mathematical sciences students, as well as a prescribed elective for students from other schools. The published aim is a careful treatment of the principal topics of Calculus I and Calculus II — the real numbers, basic topology of the real line, sequences and series of numbers and of functions — and to illustrate the power of the subject through a variety of applications.
The published framing of what changes is direct: after this course you will be able to understand better many results from Calculus I and II which you studied without proofs. That sentence is the course. The content is largely familiar; the demand is that you now prove it, and ILO 2 is stated as proving rigorously the mathematical statements in MH1100 and MH1101.
NTU's own advice to students in this outline is unusually blunt and worth reading as a difficulty signal rather than as boilerplate. It opens by saying this course is not like Calculus I and II, so get with it from the start or you will have great difficulty later, and continues that the more you work in January, the less you have to worry in April.
Always treat your own course outline and the exam timetable as authoritative.
Difficulty & time commitment
Is MH3100 hard, and how much time does it take?
MH3100 is manageable if you keep a weekly rhythm and treat the back half as the main event. The pattern is consistent: it starts gently and steepens, and the heaviest assessment is the part that separates grades.
The difficulty curve and the assessment weighting point the same way: the back half is harder and worth more. Front-loading effort there is the highest-return decision in the course.
Is this course for you
Who tends to do well, and who tends to struggle
You will likely do well if
- You start writing proofs in week 1 rather than reading them. The published advice on working early is not generic encouragement — it is specific to how this material compounds.
- You treat definitions as objects to be manipulated. Almost every proof in the course is definition unpacking followed by an estimate.
- You go to tutorials and attempt the problem sets first. The outline states feedback is delivered through the weekly problem tutorial sets.
- You summarise each theorem in your own words, which is the fourth item on NTU's own published advice list.
You may struggle if
- You wait for a worked template before attempting a proof. There is no template that survives contact with a new statement.
- You skip the epsilon arguments because you met the results in calculus and believe them already. Believing them is not what is assessed.
- You miss the midterm. There is no make-up, and 26% is more than a quarter of the grade.
- You leave sequences and series of functions to revision; weeks 11 and 12 hold the distinctions that the final examination most reliably tests.
- Prove one theorem a day from a blank page, then compare with the text. Recognition and reconstruction are different skills and only the second is examined.
- Keep a counterexample file. For each theorem, the example showing what fails when a hypothesis is dropped is worth more than a second proof of the theorem.
- Read Abbott alongside the lectures; it is written to motivate why each definition takes the form it does, which is exactly what makes proofs constructible.
- Practise writing under time. A two-hour paper on proof material rewards clean structure over completeness, and the midterm is your only rehearsal.
Syllabus
The 12 topics, topic by topic
The exam-weight marker on each topic shows where the marks concentrate. The amber topics carry the highest exam weight.
T1 · The real numbers
Weeks 1-2; ILO1-3Completeness and the least upper bound property — the axiom that separates the reals from the rationals.
T2 · Sequences: convergence and the formal definition
Week 3; ILO1-3Convergence stated precisely and proved, rather than observed.
T3 · Monotone convergence, subsequences and Cauchy sequences
Week 4; ILO1-3The three routes to proving convergence without knowing the limit in advance.
T4 · Series and convergence tests, proved
Week 5; ILO1-3The tests from Calculus II, this time with their proofs and their exact hypotheses.
T5 · Basic topology of the real numbers: open and closed sets
Week 6; ILO1-3Open, closed, limit points and closure, on the real line.
T6 · Compactness and connectedness on the real line
Week 7; ILO1-3The properties that make the extreme value and intermediate value theorems work.
T7 · Functional limits and continuity
Week 8; ILO1-3Continuity redefined topologically, in the week the midterm test falls.
T8 · Consequences of continuity
Week 9; ILO1-3Uniform continuity, and the theorems on compact sets that calculus assumed.
T9 · Derivatives
Week 10; ILO1-3The mean value theorem proved, and the results that follow from it.
T10 · Sequences of functions: pointwise and uniform convergence
Week 11; ILO1-3The distinction that makes the difference between a true and a false theorem.
T11 · Series of functions and their properties
Week 12; ILO1-3When a limit of continuous functions is continuous, and when term-by-term operations are valid.
T12 · Revision
Week 13; ILO1-3Consolidation across the whole course.
How it's assessed
Assessment structure
| Component | Weight | Format & timing |
|---|---|---|
| Final Examination | 50% | Two-hour final examination assessed against all three ILOs and against the published SPMS-MAS graduate attributes, marked to a rubric in the outline appendix. Examination period. Summative assessment. |
| Midterm test | 26% | Midterm test assessed against all three ILOs, marked to a published rubric. Absence without valid leave scores zero and no make-up is arranged. Week 8. Continuous assessment. |
| Online quiz 1 | 8% | Technology-enhanced learning quiz assessed against ILO 1, marked to a published rubric. Across the semester. Continuous assessment. |
| Online quiz 2 | 8% | Technology-enhanced learning quiz assessed against ILO 1, marked to a published rubric. Across the semester. Continuous assessment. |
| Online quiz 3 | 8% | Technology-enhanced learning quiz assessed against ILO 1, marked to a published rubric. Across the semester. Continuous assessment. |
- The five components sum to 100, and the published assessment table states a total of 100%. Late submission of assignments is not accepted. Absence from the midterm without valid leave scores zero and no make-up midterm is arranged; where the reason for absence is valid, the total course marks are rescaled to a base of 100%.
- The two-hour final examination carries 50% and is assessed against all three ILOs and the full published graduate attribute set. The 26% midterm in week 8 is mapped identically, so it is a genuine rehearsal rather than a partial check. The three online quizzes at 8% each are mapped to ILO 1 alone, which is the explanatory outcome rather than the proof outcome.
This is an exam-cram course. With the exams at 76% of the grade and the final examination alone at 50%, your result is overwhelmingly decided by how well you perform under time pressure. Summative assessment.
Final exam timing: Two-hour final examination during the examination period. Confirm the exact date and venue on your exam timetable.
How to actually pass it
A weekly rhythm, two checklists, and the traps to avoid
The course rewards consistency over cramming, and practice over re-reading. Here is the loop that works, then what to have nailed before each exam.
The weekly loop
Before the mid-semester checklist
- State and use the completeness property of the real numbers.
- Prove convergence of sequences from the formal definition, and via monotone convergence and Cauchy criteria.
- Prove the standard series convergence tests and state their exact hypotheses.
- Work with open sets, closed sets, limit points and compactness on the real line.
Before the final heaviest topics
- Prove functional limits and continuity from the definition, including uniform continuity.
- Prove the mean value theorem and derive its consequences.
- Distinguish pointwise from uniform convergence and prove which properties transfer to the limit.
- Apply methods of real analysis to problems in science and engineering involving real structures.
The mistakes that cost marks
Assuming what is to be proved. The most common failure in analysis proofs, and the hardest to see in your own work. Writing the statement you are allowed to use at the top of the page prevents most of it.
Quantifier order reversed. For every epsilon there exists a delta is not the same statement with the quantifiers swapped. Uniform continuity and uniform convergence both live on this distinction.
Calculus intuition substituted for argument. Knowing a result is true is what you arrive with. ILO 2 asks you to prove the statements you previously studied without proofs.
Midterm missed. No make-up is arranged and absence without valid leave scores zero. At 26%, that is not recoverable in the final alone.
Teaching team
Who teaches MH3100
The bios below are factual. We do not rate lecturers; any star ratings are submitted by students who have taken MH3100.
Teaching team as listed in public course information. AskSia does not rate lecturers; star ratings are submitted by students who have taken MH3100.
Formula & concept sheet
The vocabulary and formulas you must own
- Completeness
- The property that every non-empty set of reals bounded above has a least upper bound.
- Supremum
- The least upper bound of a set, which need not belong to the set.
- Cauchy sequence
- A sequence whose terms eventually become arbitrarily close to one another; in the reals, equivalent to convergence.
- Monotone convergence theorem
- A bounded monotone sequence converges, without the limit needing to be identified in advance.
- Subsequence
- A sequence formed by selecting terms of another in their original order.
- Limit point
- A point every neighbourhood of which contains a point of the set other than itself.
- Open set
- A set in which every point has a neighbourhood contained entirely within the set.
- Closed set
- A set containing all of its limit points.
- Compactness
- On the real line, the property of being closed and bounded, which makes continuous functions attain their extremes.
- Uniform continuity
- Continuity with a single tolerance valid across the whole domain rather than point by point.
- Pointwise convergence
- Convergence at each point separately, where the rate may depend on the point.
- Uniform convergence
- Convergence at a single rate valid across the whole domain, which preserves continuity in the limit.
- Mean value theorem
- On a suitable interval, some interior point has instantaneous rate of change equal to the average rate.
Common acronyms: AU · ILO.
Where it fits
Prerequisites, related courses & why it matters
The published prerequisite is MH1100 with MH1101, or CY1601, or MH1802. MH3100 carries 4 Academic Units, with 39 hours of lectures and 12 hours of tutorials, offered in Study Year 2, Semester 2, and is mutually exclusive with MH310S. The published textbook is Stephen Abbott, Understanding Analysis, 2nd edition (Springer, 2015), with Bartle and Sherbert, Introduction to Real Analysis (Wiley, 2011) as reference.
Your MH3100 study toolkit
Study the course with Sia, not just read about it
Each tool already knows MH3100: your syllabus, your texts, and where the marks are. Grouped by how you study, from first contact to exam week.
FAQ
Frequently asked questions
Is MH3100 hard?
It rates as the hardest of the NTU mathematics courses covered here after MH2220 Algebra I. NTU's own published advice to students opens by warning that the course is not like Calculus I and II and that you must get with it from the start or face great difficulty later.
What is the assessment breakdown?
A 50% two-hour final examination, a 26% midterm test in week 8, and three online quizzes at 8% each. The published table states a total of 100%.
Is there a make-up midterm?
No. The outline states that a student absent from the midterm without valid leave of absence is given zero, and that no make-up midterm will be arranged. With a valid reason for absence, the total course marks are rescaled to a base of 100%.
What is actually different from Calculus II?
The content overlaps heavily; the demand does not. ILO 2 is to prove rigorously the mathematical statements in MH1100 and MH1101. You are being assessed on constructing arguments, not on producing answers.
What do I need before taking it?
MH1100 together with MH1101, or CY1601, or MH1802. The course is mutually exclusive with MH310S.
Who coordinates the course?
The published course coordinator is Assoc Prof Chua Chek Beng, Division of Mathematical Sciences.
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