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I am new to real analysis and recently when I am reading about how $\sin(na)$ diverges as $n\rightarrow\infty$. There is a statement that I can't comprehend.

$\forall a\in (0,\pi)\exists n_k\in Z\rightarrow 2k\pi + \frac{\pi}{2}-\frac{a}{2}\leq n_ka\leq 2\pi k+\frac{\pi}{2}+\frac{a}{2}$

Is it possible to prove it as well as providing the intuition about it? I find it abstract especially when another sequence is involved.

Andes Lam
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    What exactly is it that you do not comprehend about the statement? – Ben Grossmann Jul 01 '20 at 11:53
  • This post and this post are relevant – Ben Grossmann Jul 01 '20 at 11:55
  • In other words, are you trying to understand what the statement says or are you trying to prove the statement? – Ben Grossmann Jul 01 '20 at 11:58
  • Ideally I would like to achieve both but intuition is preferred. – Andes Lam Jul 01 '20 at 11:59
  • Fine, but do you understand what the statement says or do you not? – Ben Grossmann Jul 01 '20 at 12:02
  • Partially, so when a is scaled up, it is within something. I understand $\exists k$ s.t. the range starts at $2k\pi$. But starting from $\frac{a}{2}$ and $\frac{pi}{2}$ as well as how $k$ hints to $n_k$ confuse me. – Andes Lam Jul 01 '20 at 12:04
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    I would prefer "oscillating between $-1$ and $1$" instead of "diverging". The statement states that , given a fixed arbitary small positive distance, we can find infinite many $n$ such that $na$ is within this distance to some point with $\sin(x)=1$ (and analogue to some point with $\sin(x)=-1$). This implies that the sequence won't converge. – Peter Jul 01 '20 at 12:05

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