On the modes of convergence of the series $\sum\limits_{k=1}^{\infty} \frac{1}{k} \sin \left(\frac{x}{k+1} \right)$

Question

Show that

$$f(x) = \sum_{k=1}^{\infty} \dfrac{1}{k} \sin \left(\dfrac{x}{k+1}\right).$$

converges pointwise on $\mathbb{R}$ and uniformly on each bounded interval in $\mathbb{R}$, to a differentiable function $f$ which satisifies $\vert f(x) \vert \leq \vert x \vert$ and $\vert f'(x) \vert \leq 1$.

This problem comes from a section that covers the Weierstrass M-Test and Dirichlet's Test. I am not sure where to begin with this. Analysis really isn't my thing at this point.