There is same question, but I want to check only one subtle thing from the problem.
Is $\int_1^N (n-[n])\sin(nx)\,dn$ bounded for all $N$ with $0<x<2\pi$?
(Here, $[n]$ is greatest integer function.)
It is seemingly certain, but I can't prove it.
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Do you want the integral to be bounded for each $N$, or to be uniformly bounded in both $x$ and $N$? – Sangchul Lee Jan 09 '24 at 06:05
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If we fix $N$ then $$\left| \int_1^N \big(n-[n]\big) \sin(nx) \right| \,\Bbb dn \leq \int_1^N (1 \cdot 1) \,\Bbb dn = N-1 $$ that is because $\big|\sin(nx)\big| \le 1$ and $\big|n - [n]\big| \le 1$.
Adam Wrzesiński
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Why what seems to be the problem in the last inequality? 0≤|x-[x]| <1 is always true, are trying to say he messed up in the less than equal to sign? – Ham Lemon Jan 09 '24 at 09:20