§ Diagonal lemma for monotone functions
- Statement: For a monotone function , we have the equality
- Since , by monotonicity of , we have that .
- On the other hand, note that for each , we have that . Thus each element on the RHS is dominated by some element on the LHS.
- So we must have equality of LHS and RHS.
§ Proving that powering is continuous
- We wish to prove that is continuous, given that and is continuous.
- Proof by induction. is immediate. For case :
- See that we used the diagonal lemma to convert the union over into a union over .