Show that the series $\sum_{n=1}^{\infty} f(x_n)$ converges I'm having difficulties with this exercise of calculus
Let $f:(-1, 1) \to \mathbb{R}$ be differentiable and such that $f(0)=0$ and $|f'(x)| \leq 2$ for every $x \in (-1, 1)$. If $(x_n)$ is a sequence of $(0, 1)$ such that the series $\sum_{n=1}^{\infty} x_n$ converges, show that the series $\sum_{n=1}^{\infty} f(x_n)$ converges. 
I'm sorry for my lack of effort but I really don't know where to start here. I don't know what I should make of all the information that is given in the above problem. Why does the fact that the derivative is bounded and that $f(0)=0$ is important in this problem?    

Using the mean value theorem as proposed by  Henning Makholm, one has
$$\frac{\left|f(x_n) - f(0)\right|}{\left|x_n - 0\right|} = \left|\frac{f(x_n) - f(0)}{x_n - 0}\right| = |f'(x)| \leq 2.$$
It follows that
$$|f(x_n)| \leq 2|x_n|.$$
Since $x_n \in (0, 1)$ for every $n \in \mathbb{N}$, we can remove the absolute value on the RHS so that 
$$|f(x_n)| \leq 2 x_n.$$
Therefore 
$$\sum_{n=1}^{\infty} |f(x_n)| \leq 2\sum_{n=1}^{\infty} x_n$$
and because the series on the right converges by assumption, $\sum_{n=1}^{\infty} f(x_n)$
must also converges by the comparison test. 

Can someone verify if my work is correct?
