0.3: Proof Do's and Dont's

Do's:

1. Write the statement to be proved. It should be clear what you are proving.
2. Clearly mark the beginning of your proof with the word "Proof".
3. Make your proof self-contained. In particular, identify all variables used in your proof in the body of your proof.
4. Write proof in complete English sentences.

Example $\PageIndex{1}$: Acceptable

Proof: Let $n\in \mathbb{Z}$. Assume $n$ is an even integer. Then $n=2k,$ for some $k \in \mathbb{Z}$.

Example $\PageIndex{2}$: Unacceptable

Proof: $n$ is even $\implies$ 2k.

5. Indicate what method of proof you are using. (The default assumption is that it is a direct proof).

6. Learn the definitions and how they come into play when proving various types of statements.

Don'ts:

1. Argue from examples. A general statement can't be proved true by showing it is true for special cases.
2. Use the same letter to mean two different things within a proof.

Example $\PageIndex{3}$:

Proof: Let $n\in \mathbb{Z}$. Assume $n$ is an even integer. Then $n=2k,$ for some $k \in \mathbb{Z}$. So $n^2=4K^2=2(2k^2)$. Thus $n^2=2k, k \in \mathbb{Z}$ is even. (The reader asks, does $n=n^2$?)

3. Assume what you are trying to prove. This is also known as begging the question. You can do it inadvertently in the middle of proof if you are not careful.