Yes, I also think that the question is strange, but I can't rephrase it.
I need to find $\inf(X)$ and $\sup(X)$ if $X = \{X_n, X_n = 3 \sin 4n, n\in\mathbb{N}\}$.
I'm sorry, but I don't even know how to start the process of finding the solution. If you can give me the hint I will be very pleased.
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Nigel Overmars
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Mex
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1My english is bad, I know. – Mex Sep 16 '15 at 15:14
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Hint: The set $X(q) = \{\sin nq \, | n \in \mathbb{N} \}$ is dense on the interval $[-1,1]$ if $q \neq 0$ is rational. This follows from the irrationality of $\pi$ and the periodicity of $\sin$.
user2566092
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@Mex Yes although you'll never actually achieve those values in your sequence, just get closer and closer to them. – user2566092 Sep 16 '15 at 15:25
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2@Mex I've left out a lot of details but you asked for a hint so hopefully this is good enough. For a possible proof, try considering $[\pi/2, \pi/2 + 1/m]$ and $[3\pi/2, 3\pi/2 + 1/m]$ and showing you must get at least some values of $4n$ that fall within these intervals when $m$ is any positive integer and you compute the angle after subtracting $2 \pi k$ for appropriate $k$ to get the angle value of $4n$ between $0$ and $2 \pi$. – user2566092 Sep 16 '15 at 15:31