Proving some trigonometric series are convergent

Question

I have to prove that if $(a_k)_k$ is decreasing and tending to 0, then, for all $\epsilon>0$, the series $\sum a_k \sin(2\pi kx), \sum a_k \cos(2\pi kx)$ and $\sum a_k e^{2\pi ikx}$ are all uniformly convergent on $(\epsilon,1-\epsilon)$. I have to use the fact that if $(v_k)_k$ is positive and decrasing and $(u_k)_k$ an arbitrary sucession, then $$ |u_1v_1+...+u_nv_n| \leq v_1\max_n \left|\sum_{k=1}^n u_k\right| $$ I have seen a proof using that $$\sum_{k=1}^n \sin(2\pi kx)=\csc(\pi x)\sin(N\pi x)\sin((N+1)\pi x)$$ but I would like to know how to prove that equality because I can't see it. Thanks in advance.