Problem: Suppose $f$ is defined and differentiable for $x>0$, and $f'(x)\rightarrow 0$ as $x\rightarrow +\infty$. Put $g(x)=f(x+1)-f(x)$. Prove that $g(x)\rightarrow 0$ as $x\rightarrow +\infty$.
Source: W. Rudin, Principles of Mathematical Analysis, Chapter 5, exercise 5.