Prove Theorem 3.1.

For any two functions and , we have if and only if and .

To prove this theorem, we need to show the logic holds both ways.

If , then for . As for , . And as for , .


Now we need to prove the other way around.

If , then for . And if , then for .

Combining the above two inequalities, we can say for , i.e. .

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