Given a twice differentiable function $f(x)$ on $\mathbb R$ with the following properties:
$f$ is an increasing function in $\mathbb R$
There is a sequence of real numbers $\{x_{n}\}_{n=-\infty}^{n=\infty}$, and a constant $c>0$ such that $f(x_{n+1})-f(x_n)=c$ for all $n$.
(Edit: $\lim\limits_{n \to -\infty} x_{n}=-\infty$)
Now, is it true that \lim \limits_{x \to -\infty} f'(x) cannot be zero?
I think it is true, because if \lim\limits_{x \to -\infty} f'(x)=0 then the slope of the tangent line at $-\infty$ will be very close (approach) to zero and this means--since $f$ is increasing--that $\lim\limits_{x \to -\infty}f(x)=a $, for some constant $a$, this can be seen by a graph!. This means that the assumption is not correct because of item #2 above. Please correct me if my argument is not right!
Maybe my proof is not correct, but what about the problem itself!