Step by step
The question
Question: prove that there is no largest even number.
1 Assume the opposite
Suppose the opposite: there is a largest even number.
Call it N.
Suppose N is the largest even 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 even number too: even number numbers come every 2.
N + 2 is an even 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 even number than N ✗
The opposite led to nonsense, so there is no largest even number.
So there is no largest even number. ∎
4 Check
Try a big one: 1000.
1002 is even and bigger.
Right!
N = 1000: 1000 + 2 = 1002 > 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 even number. ∎