Solution
= Solution
It is not uniform. For a small fixed $c\gt 0$, set $x_N=c/N^3$. On a block $N\le n\le(1+\delta)N$, all $n^3x_N$ lie in a compact subinterval of $(0,\pi)$, so the corresponding tail has a positive lower bound independent of $N$; this violates the uniform Cauchy criterion.