Step by step
The question
Question: prove that there is no largest odd number.
1 Assume the opposite
Suppose the opposite: there is a largest odd number.
Call it N.
Suppose N is the largest odd number.
Careful!
A huge number is still just one example.
We must show it for every N.
Careful: a big example is not a proof
2 Work it out
Look at N plus 2.
It is odd number too: odd number numbers come every 2.
N + 2 is an odd number
And N plus 2 is bigger than N.
N + 2 > N
3 Find the contradiction
So N was not the largest after all.
That is nonsense!
N + 2 is a bigger odd number than N ✗
The opposite led to nonsense, so there is no largest odd number.
So there is no largest odd number. ∎
4 Check
Try a big one: 1001.
1003 is odd and bigger.
Right!
N = 1001: 1001 + 2 = 1003 > 1001
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer
So there is no largest odd number. ∎