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17Calculus - Proofs

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After calculus and differential equations, the study of mathematics takes a dramatic turn from calculations to proofs. You need to learn some general things about proofs, as well as how to read and understand them before you can begin to learn to write them. Additionally, as an undergrad, you will understand calculus a lot better if you understand proofs. This is especially important if you are a math major.

List of Math Proofs

Here is a list of math proofs on 17calculus as well as links to proofs on other sites. Some of these proofs are written out, others are in videos.

Infinite Series

p-Series Convergence Theorem Proof


\(\displaystyle{ \lim_{\theta \to 0}{\frac{\sin(\theta)}{\theta}} = 1 }\)

[link to proof]

Getting Started

When learning math, you don't want to skip anything, including theorems and proofs. Theorems tell you under what conditions the results can be applied and proofs show you how to use the math you already know to produce new uses and applications. Here are some suggestions to help you read and understand math proofs. At the end of this page, we recommend several books that you may want to consider if you want a deeper understanding of proofs.

Let me say up front . . . most students just read theorem statements and skip proofs. If you work through each theorem statement and study the proof, you will begin to actually understand math and know when to use it. This will put you ahead in your class and in future classes. Some instructors will also give problems on homework and exams that the other students will miss because they didn't know when to apply which theorem. But you will get it right. This could potentially increase your grade. So, let's get to it.

How do you actually study proofs and theorems so that you can understand them? As you read on the page about how to read math books, you can't just read theorems and proofs like you would read a novel. Mathematicians write theorems and proofs as elegantly as they can and this will usually obscure what is going on. Even Ph.D.'s have to work through theorems and proofs, so you are in good company.

We will separate our discussion into two parts, the theorem statement and the proof. Start by getting out a pencil and several pieces of paper.

Understanding The Theorem Statement

Theorems are written in concise, very compact language which inevitably obscures what is being said. So you need to dig out the pieces in order to understand them. Theorem statements have two main parts, the conditions and the conclusions. The conditions tell you what has to be true in order to apply the theorem. The conclusions are what you can know for sure is true as long as all the conditions hold. The nice thing about theorem statements is every single condition that is required will be stated somewhere in the theorem. So you don't have to guess about a condition.

Your main task is to separate out the conditions and the conclusions. If the language of the theorem (like English) is a language you are very familiar with, then this will not be hard. Mathematicians are very concise, so every word is important. Do not skip any word, no matter how small.

At this point, it won't help you understand what to do if we just give you a bunch of generic rules to handle theorems. So we will use an example to show you how to do this, from which you should be able to extrapolate what you need to do. Here is a theorem from differential equations. You should know some math but don't worry if you don't understand all the advanced math, that's not the point. The point is to extract the information we need.



If the functions \(p\) and \(g\) are continuous on an open interval \(I: \alpha < t < \beta\) containing the point \(t=t_0\), then there exists a unique function \(y=\phi(t)\) that satisfies the differential equation \(y'+p(t)y=g(t)\) for each \(t\) in \(I\), and that also satisfies the initial condition \(y(t_0)=y_0\) where \(y_0\) is an arbitrary prescribed initial value.

This is a theorem from differential equations that determines under what conditions a solution exists and is unique, i.e. it is the one and only solution.

Almost all theorems have an if-then format, where the if and/or then may not be explicitly stated but the structure of the sentences will tell you which section is which. In the theorem above, we do have if-then sections. So, first, we write down all the conditions covered by the if section.


\(p\) and \(g\) are functions which are continuous

\(p\) and \(g\) don't have to be continuous everywhere, just on the open interval \(I\)

\(I\) is an open interval \(\alpha < t < \beta\)

\(I\) must contain the point \(t=t_0\)

Although not stated up front, \(p\) and \(g\) are functions of t and can be written \(p(t)\) and \(g(t)\). You can see this from the differential equation \(y'+p(t)y=g(t)\).


1. a solution is guaranteed to exist

2. the solution is unique, i.e. there is only one

They actually name the solution \(y=\phi(t)\). This name may be used in the proof or the subsequent discussion in the text but it doesn't come into play in the theorem statement.

Understanding The Proof

Most proofs you find in textbooks and other books are deliberately condensed into a concise format and language that mathematicians use. In order to understand a proof, you need to go through each word and sentence and fill in the blanks until the proof flows in an understandable manner to you. Rewrite the proof with extra notes that help you understand it. Proofs contain all the information needed to prove the statement. However, not everything is obvious. So you need to rewrite it and review it multiple times in order to understand it.

Memorizing The Proof

This section is based on personal experience. In the beginning grad math class where we learned about proofs, my instructor would write out a proof on the board but not allow us to take notes. He would give us about 10 minutes to read and memorize the proof. Then our assignment for the next class period was to write out the proof and turn it in. He would correct the proof and return it to us. Then one or more of the proofs would appear on the next exam. We had to memorize at least 10-15 proofs for each exam. By the end of the semester, we had to be able to reproduce 50-75 proofs.
At first, it was very difficult but as the semester went on, it became easier and easier to understand and memorize proofs. I found that it was just not possible to memorize a proof without understanding it. I had to understand how each step leads to the next. It was an extremely effective technique, while at the same time, it was very unpopular.

I mention it here since I have not seen the same technique taught since. I believe you would benefit from this technique if you tried it on your own. See the article on memorizing to get ideas on how to help you.

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