Question 1
Given natural numbers $N,n$, prove that for every nonnegative real number $\alpha\le N$ and every real $x$,
$$\left|\sum_{k=0}^n\frac{\sin((\alpha+k)x)}{N+k}\right|\le\min\left\{(n+1)|x|,\frac1{N|\sin(x/2)|}\right\}.$$
$$\left|\sum_{k=0}^n\frac{\sin((\alpha+k)x)}{N+k}\right|\le\min\left\{(n+1)|x|,\frac1{N|\sin(x/2)|}\right\}.$$