Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function. Prove that if $f$ is a function such that $f'(x) > 0$ for $x \in \mathbb{R}$ then $f$ is a one to one function.

Set $f(x)$ to be some function such that $f'(x) > 0 \implies$ f(x) is increasing $\forall x$. Take two arbitrary and distinct values $a$ and $b$ with $a < b \implies f(a) < f(b) \implies \nexists a, b : f(a) = f(b) \therefore f$ is a one to one function.

Is this a sufficient proof?

Looks good. ${}{}{}{}{}{}{}{}{}{}{}$
