Prove that the set of increasing sequences of natural numbers (n1<n2<...) is uncountable.
Suppose we can enumerate them, for example,
...
Define a new sequence:
I hope you can see how we defined this sequence? The first term is the boxed term +1, then the second term is the last term plus the second boxed term, and so on. This creates a new increasing sequence which is not found on this list.
(This was the diagnol argument)
EDIT: Mistake fixed.