Find $\inf(X)$ and $\sup(X)$ then check by definition of supremum and infimum.

Question

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.

Answer 1

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 $\sin$.