This is a typical proof one may see in a undergraduate introduction to proofs mathematics course.
Note are the Fibonacci numbers where and where .
Assume is rational; . This implies
and are fixed integers. Next, lets consider the difference for
Note, the difference is greater than zero:
Now, we can estimate this difference. From the definition of , we can see that
So, we have arrived at the following bounds
Let . We obtain
Multiplying the inequality by we obtain
Note that and so this further restricts the bounds as
Since we can pick large enough such that
This is a contradiction and thus our assumption must be wrong. must be irrational; .
And that’s it! How would you have proven this?