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?