Book: Mathematical Analysis by Apostol
Theorem 2.2: .
Let . Prove that if
My attempt: Let Then , by assumption. Then, such that . Similarly, such that .
Since , , but . Then . Since , , but .
So we have that and . Since and , .
This is equivalent to saying that for and , we have that .
By theorem 2.2, it must be that . So implies that . Therefore, f is injective.
Have I made any errors in my argument?