Let $f: \mathbb R \to (E,||\cdot||)$, where $(E,||\cdot||)$ is a normed linear space. Suppose that $f$ is differentiable on $\mathbb R$ and that $\lim_{x \to \infty} f'(x) = 0$. Prove that $\lim_{x \to \infty} f(x+1) - f(x) = 0$.
In the simple case $E = \mathbb R$, this can be shown by MVT. In some other particular cases I can prove this as well. But I cannot see how this might be done in an arbitrary $E$: I do not have any relevant theorems as tools.
Any hints or suggestions?