From Calculus to Real Analysis

This page is intended to be a bridge between the Calculus taught in high-school and a Real Analysis course at the undergraduate level. It is assumed that the reader is a high-school student, who wants to learn Calculus rigorously, but gets intimidated by the level of sophistication in any UG level book on Real Analysis. This is why I shamelessly tried to write some notes on my own, on some specific topics. I must admit that sometimes statements are loosely written and due to lack of experience in teaching, naivety is present everywhere. Nevertheless, I hope it would serve the purpose  mentioned above. The notes may contain some errors (typo’s), be skeptical while reading them.

A. Sequences and Series

  1. Sequences and Limits
  2. Subsequences Correction: On page 3 (proof of result 3), the ordering should be corrected to: x_{\ell_1}< x_{\ell_2} <\dots
  3. Monotone Sequences, Solutions
  4. Sandwich Theorem, Solutions
  5. Cauchy Sequence: Notes, A classnote, Solutions
  6. Series of real numbers:
    • Classnote (from an online class)
      A correction: An alternating series \sum_{n\ge 1} (-1)^n a_n where a_n>0 for all n, converges if a_n decreases to 0 (just \lim_{n\to\infty} a_n=0 is not sufficient).
    • Problems, Solutions Correction in #20 part (b): the RHS should read n(2n-1)/3.
    • Notes by J. Hunter: First part (that you must learn), some advanced things (optional at this level, but you should read if you are interested).

Class Test 1 (2019): Question Paper, Solution.
Class Test 1 (2020): Question Paper (as a Google form)

\star More problems on Sequences and Series

B. Continuity and Limit of a function

Those who don’t have a good grip on functions may read this and this. Here are some problems on functions.

  1. Definitions of Continuity (with problems and solutions)
  2. A Tale of Rationals and Irrationals (with problems and solutions)
  3. Theorems on Continuity (note + problems), Solutions
  4. Limit of a function:

Class Test 2 (2019): Question paper, Solution.
Class Test 3 (2019): Question Paper, Solution.
Class Test 2 (2020): Question paper, Solution, discussions.

C. Differentiation

Some references:
Differentiable Functions (written by John Hunter)
Limit, Continuity,  and Differentiability (by M. T. Nair)

  1. Classnote of an online class
  2. Introductory Problems (Solutions)
  3. Fermat’s theorem and the Mean Value Theorems
  4. Application of Derivatives Results, Problems

Class test 4 (2019): Question Paper
Class test 3 (2020): Question Paper, Solutions

D. Integration

  1. Integration Day1 Notes (with exercises), Solutions to Exc1
  2. Integration Day2 Notes (with exercises), Solutions to Exc2
  3. Integration Day3 Notes (with exercises), Solutions to Exc3
  4. Integration Day4 Notes

Integration compiled notes: click here to download. (Many thanks to Srijon Sarkar for pointing out numerous typos.)

Bonus Topics

  1. Uniform Continuity
  2. L’Hôpital’s rule: notes, problems, solutions (to selected problems)
  3. Convexity: problems, classnote 
  4. Taylor’s theorem: Note by John Hunter, Another note
  5. Several routes connecting to the number e
  6. A simple proof that pi is irrational (link to the paper by Ivan Niven)
  7. Stirling’s formula for n!

More References

  1. Introduction to Real Analysis by Bartle and Sherbert can serve as a good textbook. The book is for undergraduates, so you need not read each and every section of the book.
  2. If you can read Bengali, try Arnab Chakraborty’s Real Analysis books (3 books). They can be bought from bookstores in College Street, Kolkata. Also found at Amazon. These books are fantastic and most student-friendly!
  3. A collection of Analysis notes by John Hunter can be found on this website. I found these notes to be really great! However, some parts of these notes might be very difficult even for a brilliant high school student; save those parts for your undergraduate years in the future!
  4. Here is another nice book in an online format that you can read: Calculus for Beginners and Artists. It is at the basic level, but focus more on real life applications.
  5. The following blog by Yves Simon contains many worked out problems: https://alteredzine.blogspot.com. If interested, you can also check out his Calculus courses.

Two Amazing Playlists from YouTube

The following is a playlist from MIT OpenCourseWare, titled “Big Picture of Calculus”, given by Prof. Gilbert Strang.

The second playlist, titled “The Essence of Calculus”, is from the channel 3blue1brown.