What statements to write for using proof by cases in a proof by contradiction?

Question

In a theorem that I am trying to prove I first use proof by contradiction and for reaching contradiction I need to prove using cases.

Now for formally writing proof by contradiction we use the following structure -

Method: In order to prove a proposition P by contradiction:

1. Write, “We use proof by contradiction.”

2. Write, “Suppose P is false.”

3. Deduce something known to be false (a logical contradiction).

4. Write, “This is a contradiction. Therefore, P must be true.”

I have the doubt that how do I incorporate a statement that I will use proof by cases when I actually am not going to prove the statement true but want to reach a contradiction in each of the cases.

I have figured out the proof but do not understand how to present it correctly.

Answer 1

Your case analysis is entirely within step 3 in your plan, so you should write that as part of that deduction like you would write any other case analysis. A possible phrasing could be

Theorem. Assume that the snark is a boojum. Then the florbax is green.

Proof. Suppose, for a contradiction, that the florbax is not green. Then, by lemma 1.3.4.4.2, it must be either red or blue, and we divide into cases based on that.

1. If the florbax is red, then bla bla bla bla bla, which is a contradiction.

2. If the florbax is blue, then foo bar baz quux gargle, which is a contradiction.

Since a contradiction results no matter whether the florbax is red or blue, it must be green.

The last line can often be left implicit, except if (a) the cases are so long that there's a risk that the reader will have lost the thread and you need to remind them how it all fits together, or (b) you're doing classwork where you're specifically expected to say certain magic phrases during the proof, as a way to check that you understand the logical structure of your proof.