If Real Analysis I was about understanding the “ground rules” of the real number line, Real Analysis II is where you build the skyscrapers. It’s the transition from basic sequences to the heavy lifting of Riemann-Stieltjes integrals, sequences of functions, and the intricate world of Lebesgue measure.
Below is the past paper download link:
Download Real Analysis II Past Paper for Revision
Above is the past paper download link:
At this level, “understanding” the lecture is only half the battle. The real test is whether you can replicate a multi-step epsilon-delta proof or justify the interchange of a limit and an integral under exam conditions. This is where past papers become your most valuable asset.
FAQ: Mastering the Rigor of Real Analysis II
Q: Real Analysis II feels more like “logic” than “math.” How do past papers help with this?
A: You’re right—it is logic. Most students lose marks not because they can’t calculate, but because their proofs aren’t rigorous enough. By downloading a past paper, you see the exact standard of proof required. You learn to stop saying “it’s obvious that…” and start using “by the Weierstrass M-Test, it follows that…” Past papers teach you the vocabulary of a mathematician.
Q: What are the high-yield topics I should focus on in this specific module?
A: Real Analysis II usually shifts focus toward the behavior of functions and integration. Look for these key areas in your practice:
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Uniform vs. Pointwise Convergence: Can you prove that a sequence of continuous functions converges to a function that is also continuous?
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Riemann-Stieltjes Integration: Moving beyond the standard Riemann integral to handle more general cases.
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The Fundamental Theorem of Calculus: Expect to prove it, not just use it.
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Power Series: Determining the radius and interval of convergence using the Ratio or Root tests.
Q: I get overwhelmed by the Epsilon ($\epsilon$) and Delta ($\delta$) definitions. Any advice?
A: Use the past paper to practice “backward scratchwork.” In Real Analysis, you often start with the conclusion ($|f_n(x) – f(x)| < \epsilon$) and work backward to find your $N$ or $\delta$. When you practice with old exam questions, you’ll notice that many proofs follow the same structural “skeleton.” Once you master the skeleton, you just swap out the functions.
Q: How do I handle questions about “Measure Theory” or “Lebesgue Integrals”?
A: If your course includes Lebesgue measure, the past paper will likely ask you to compare it to Riemann integration. Practice identifying “sets of measure zero” (like the rational numbers). This is a classic exam favorite because it tests whether you truly understand why we need more advanced integration techniques in the first place.
Secure Your A-Grade Today
The leap from “knowing” a theorem to “proving” it under a 3-hour time limit is huge. Don’t let the exam be the first time you try to prove the Arzelà-Ascoli Theorem from scratch. Download our curated revision pack and start refining your logic today.
Final Tip: In Real Analysis, your definitions are your tools. If you can’t define “Uniform Continuity” perfectly from memory, you can’t prove it. Write out your definitions ten times each before you start the paper!