Sinc-squared summation of a sequence tending to zero ($k{\sum }_{r\ge 1}{A}_r{\mathrm{sinc}}^2(rk)\to 0$)

If $A_r\to 0$, then for nonzero $k$ the weighted series is absolutely convergent and its tail after $n$ is bounded by $2{\sup }_{r>n}|A_r|/(n{k}^2)$. This follows from $|\sin (rk)|\le 1$ and the reciprocal-square tail bound. For $N=\lfloor 1/|k|\rfloor$, the prefix multiplied by $|k|$ tends to zero by splitting at a fixed index where $|A_r|$ becomes small. The multiplied tail is bounded by $2{\sup }_{r>N}|A_r|/(N|k|)$, which tends to zero. Thus the whole multiplied series tends to zero as $k\to 0$. Omitting $k^{-2}$ from the unmultiplied tail bound is invalid.