Showing that a functions derivative is not bounded on $\mathbb{R}$ Suppose that $f$ is differentiable but not uniformly continuous on $\mathbb{R}$. Prove that $|f'|$ is not bounded on  $\mathbb{R}$. 
So I know that to show that $|f'|$ isn't bounded you would have to show that for any constant B, $|f'(a)|$ $>$ $B$, for some $a$ $\in$ $\mathbb{R}$. How would I go about doing this?

Hint: $A\Rightarrow B \iff \neg B \Rightarrow \neg A$. 
Assume that for $f:\mathbb R\rightarrow\mathbb R$ differentiable and $|f'|$ bounded it follows that $f$ is uniformly continuous (to be more precise, it is Lipschitz continuous).
