Let $f$ be differentiable on $\left[ a,\infty \right)$. Prove that if $\exists m>0\,\forall x\in \left[ a,\infty \right)\,f'\left( x \right)\ge m~$, then $\lim\limits_{x\to\infty}\,f\left( x \right)=\infty $.
I began by using the average value theorem (Lagrange's theorem) to prove that $f$ is monotonously increasing, however I still need to prove that $f$ is not bounded, to reach the conclusion that it diverges to infinity, but am not sure how to proceed on that.
By the mean value theorem, for each $x\in (a,\infty)$, there exists $c_x \in (a,x)$ such that $f(x) = f(a) + f'(c_x)(x-a)$. Since $f'(t) \ge m$ for all $t\in [a,\infty)$, we have $f(x) \ge f(a) + m(x-a)$ for all $x\in [a,\infty)$. Since $f(a) + m(x-a) \to \infty$ as $x\to \infty$, it follows that $f(x) \to \infty$ as $x\to \infty$.
Suppose $f(x) \le M < \infty$ for $x \ge a$. Then for all $x$ there exists some $\xi_x$ such that $f(x)-f(0) = f'(\xi_x)x$, that is $f'(\xi_x) \le {f(x)-f(0) \over x} \le {M-f(0) \over x} $. Then $\inf_{x \ge a} f'(x) \le 0$.
$$ f(n+1)-f(n) = \frac{f(n+1)-f(n)}{(n+1)-n} = f'(c_n) \ge m $$ for some $c_n$ between $n$ and $n+1$.
Prove by induction that it follows that $f(n+k)-f(n) \ge km$. Then let $k\to\infty$.
