Uniform bound for harmonic sine polynomials (${\sup }_m{\Vert {\sum }_{k=1}^m\frac{\sin kt}{k}\Vert }_{\infty }\le 1+\pi$)

For $0<t\le \pi$, split the sum at $\lfloor 1/t\rfloor$. The first part is bounded using $|\sin kt|\le kt$; summation by parts and the geometric bound on interval sine sums control the rest. The harmonic coefficient sum grows like $\log m$, but cancellations keep the whole trigonometric polynomial uniformly bounded. Modulating this polynomial lets a Fourier partial sum expose the large uncancelled half.