Knowledge of proofs and proof-writing before studying spivak's or apostol's Calculus

Question

I want to start reading spivak or apostol calculus. But I believe a certain level of proof knowledge is essential to solve the problems.

Do these books teach you how to prove results or do we learn this separately?

Answer 1

You will receive some guidance from the Spivak and Apostol, in the sense that for most proofs you'll need to write, and most proof strategies you'll need to use, you can model the abundant proofs provided by each of those authors. However, a basic familiarity with proof methods will enhance your understanding of the proofs provided by the author, and such a familiarity will greatly ease the challenges of writing proofs as required for many of the exercises.

Often proof-based courses (courses whose purpose is to teach basic logic and gain practice in proof-reading and proof-writing) are offered in the form of a "bridge" course: taken by students after having taken calculus, but before more advanced coursework in analysis or algebra. However, many universities do not offer any such bridge course, with the expectation that students acquire proficiency in reading and writing proofs while tackling increasingly abstract coursework.

For a recommendation: I'd suggest that you obtain the text: How to Prove It: A Structured Approach by Daniel Velleman. It's an excellent primer for the logic of proofs and guide to both reading and writing a variety proof types if you have time to peruse it before embarking on either of the texts you plan to study. (See the table of contents.) Even if you have only time enough to cover propositional and predicate logic, and perhaps to skim for an overview of proof strategies, it will be well worth it. In any case, Velleman's text will serve nicely as a companion text, or as a reference to consult while working through Spivak and/or Apostol.

Answer 2

Some free material recommendations:

If you are looking for a book about learning to do proofs, I recommend The Book of Proof.

This online course is pretty good, too.

However, I don't recommend reading a "proof book" before starting on Spivak or Apostol. I fell into this trap a bit myself, and when I started my proof based courses I did not feel like they helped much. What helped was to work lots of problems and have people critique my work... revise my work (and repeat). This is a perfect place to do this! In other words, I think you should just do it, make mistakes, and learn from them.

Dive in, look at a book for reference if you need to (or a short web article), and get feedback!

EDIT: Spivak and Apostol definitely show you enough to do the problems (I think Apostol more so). You might not learn terms such as "direct proof," "proof by contrapositive," etc. but you will see how to do proofs by reading the ones in the text. But you will definitely learn how to do a proof by induction, for instance.