If Real Analysis I was about laying the foundation stones of the number line, Real Analysis II is where we build the skyscraper. This is the unit where “intuitive” calculus is stripped away and replaced by absolute, rigorous proof. We move beyond simple derivatives to explore the mechanics of the Riemann Integral, the delicate convergence of function sequences, and the abstract beauty of metric spaces.
Below is the exam paper download link
PDF Past Paper On Real Analysis II For Revision
Above is the exam paper download link
For many students, this is the most “mathematical” math gets. It requires a level of precision where a single misplaced quantifier can change the entire meaning of a proof. To help you navigate this transition from calculation to high-level theory, we have compiled a diagnostic Q&A based on the most common hurdles found in recent examination papers.
Why Real Analysis II is the Ultimate Test of Rigor
In Real Analysis II, we no longer take the “Area under a curve” for granted. We ask: What does it actually mean for a function to be integrable? This unit provides the tools to handle the infinite and the infinitesimal with surgical precision. Whether you are aiming for a career in quantitative finance, theoretical physics, or pure mathematics research, the logical discipline you gain here is your most valuable asset.
Key Revision Questions and Answers
Q1: What is the Riemann Criterion for Integrability?
A: A bounded function $f$ is Riemann integrable on an interval $[a, b]$ if and only if, for every $\epsilon > 0$, there exists a partition $P$ such that the difference between the Upper Sum $U(P, f)$ and the Lower Sum $L(P, f)$ is less than $\epsilon$. In simpler terms, if you can squeeze the upper and lower rectangles close enough together that the gap between them vanishes, the function has a definitive integral.
Q2: How does Uniform Convergence protect the properties of a sequence of functions?
A: This is a classic exam focus. Pointwise convergence is “weak”—it can turn a sequence of continuous functions into a discontinuous limit. Uniform Convergence is the “strong” version. If a sequence of continuous functions converges uniformly, the limit function is guaranteed to be continuous. Furthermore, it allows you to swap the order of limits and integrals, which is a vital move in complex proofs.
Q3: What is the “Radius of Convergence” for a Power Series?
A: A power series $\sum a_n (x-c)^n$ behaves like a circle of influence. Inside the radius $R$, the series converges absolutely and behaves beautifully (you can even differentiate it term-by-term). Outside $R$, it diverges into infinity. At the boundary—the actual edge of the radius—the behavior is unpredictable and usually requires a specific test like the Ratio Test or Root Test to solve.
Q4: Explain the concept of a “Metric Space” in abstract analysis.
A: Real Analysis II often introduces the idea that “distance” doesn’t just apply to straight lines on a graph. A metric space is any set where we have a function (a metric) that defines the distance between two points. It must follow three rules: the distance is always non-negative, the distance from A to B is the same as B to A, and it must satisfy the Triangle Inequality (the direct path is always the shortest).
Q5: What is the significance of the “Equicontinuity” in the Arzelà-Ascoli Theorem?
A: This is a high-level concept often used in the final questions of a paper. Equicontinuity means that an entire family of functions is “equally” continuous—they all share the same $\delta$ for a given $\epsilon$. The Arzelà-Ascoli theorem tells us that if a sequence of functions is uniformly bounded and equicontinuous, we can always find a subsequence that converges uniformly. It is the “Bolzano-Weierstrass” theorem for functions.
How to Use This Past Paper for Your Revision
Real Analysis is not a subject you can “skim.” Once you Download PDF Past Paper On Real Analysis II For Revision, you must treat every proof as a construction project. Don’t just read the theorem; try to break it. Ask yourself: What happens if the interval isn’t closed? What if the function isn’t bounded? When practicing, focus on the “Epsilon-N” proofs for sequences of functions. Writing these out by hand is the only way to develop the muscle memory required for the exam. By the time you finish this past paper, the abstract symbols will start to feel like a second language.
Last updated on: March 21, 2026