 
Summary: WRITING PROOFS
Christopher Heil
Georgia Institute of Technology
A "theorem" is just a statement of fact. A "proof" of the theorem is a logical explanation of why
the theorem is true.
Many theorems have this form:
Theorem I. If statement A is true then statement B is true.
This just means that whenever statement A is valid, then statement B must be valid as well. A
proof is an explanation of WHY statement B must be true whenever statement A is true.
1. Direct Proofs.
There are several ways to write a proof of the theorem "If statement A is true then statement B
is true." We'll discuss several of them in these pages. It may not be obvious at first which variety of
proof to use, but a good rule of thumb is to try a direct proof first.
A direct proof. Start by assuming that statement A is true. After all, if statement A is false then
there's nothing to worry about; it doesn't matter then whether B is true or false. So, suppose that
statement A is truewrite that down as the first step. This is information that you can use and build
on. Now try to proceed logically, one step at a time, building on this information until you have shown
that statement B is true.
An important point is that a proof is always written in English! There are mathematical symbols in
with the words, but you must write clear, complete, English sentences, one after another until you've
