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Let $A_n =\sum \limits_{j=1}^n\frac{\sin j}{j}$. How can I prove that $A_n$ converges as $n$ approaches infinity?

I know that if we think of it as an infinite series, it can be proved by applying appropriate convergence tests(for example, Abel's convergence test(or Dirichlet test). But I want a proof not using convergence tests of an infinite series, and think of this as a 'sequence' $A_n$ instead of a 'series'.

I read Proving that the sequence $F_{n}(x)=\sum\limits_{k=1}^{n} \frac{\sin{kx}}{k}$ is boundedly convergent on $\mathbb{R}$ and other answers related to this question, but most of them used convergence tests and I don't know about Riemann-Lebesgue lemma or Fourier series. Actually I am taking an elementary analysis course, and this is an exercise on the textbook's 'sequence' part, not 'series'.

Arctic Char
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Sphere
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    The sequence is defined as a serie, I think every method will just be a reformulation of convergence tests in some way or using Riemann-Lebesgue lemma.. – nicomezi Sep 23 '20 at 13:25
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    Does this answer count as "not using series"? – Arctic Char Sep 23 '20 at 13:27
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    Try to prove it standing on your left leg, with your hands tied behind your back, if that's how you understand mathematics. –  Sep 23 '20 at 13:33
  • Thanks for all. I was able to modify a proof using convergence tests into something that uses sequences(especially, I proved that $A_n$ is a cauchy sequence). – Sphere Sep 25 '20 at 01:55

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