Step by step
The question
Question: prove that there is no largest integer.
1 Assume the opposite
Suppose the opposite: there is a largest integer.
Call it N.
Suppose N is the largest integer.
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 1.
It is an integer too.
N + 1 is an integer
And N plus 1 is bigger than N.
N + 1 > N
3 Find the contradiction
So N was not the largest after all.
That is nonsense!
N + 1 is a bigger integer than N ✗
The opposite led to nonsense, so there is no largest integer.
So there is no largest integer. ∎
4 Check
Try a big one: 1000.
1001 is an integer and bigger.
Right!
N = 1000: 1000 + 1 = 1001 > 1000
Let's look back.
First: assume the opposite.
Then: work it out.
Then: find the contradiction.
Last: check.
Answer
So there is no largest integer. ∎