Which of the following subsets of the real line is compact?

Question

Put $\theta:= \frac{\pi}{2019}$ and let $\mathbb{N}$ denote the set of positive integers. Which of the following subsets of the real line is compact?

(a) $\{\frac{sin(n\theta)}{n}| n \in \mathbb{N}\}$

(b) $\{\frac{cos(n\theta)}{n}| n \in \mathbb{N}\}$

(c) $\{\frac{tan(n\theta)}{n}| n \in \mathbb{N}\}$

(d) None of the above.

Definition of compact set: A set K ⊆ R is compact if every sequence in K has a sub-sequence that converges to a limit that is also in K.

Characterization of Compactness in R Theorem: A set K ⊆ R is compact if and only if it is closed and bounded.

Here I am unable to find the closed and boundedness property. For e.g.: $\{\frac{sin(n\theta)}{n}| n \in \mathbb{N}\}$ and $\{\frac{cos(n\theta)}{n}| n \in \mathbb{N}\}$ are bounded, but I am unable to determine whether closed or not.

Thanks in advance.

Answer 1

Hint: The sequences in a), b) and c) all have $0$ as the only limit point. So they are closed iff $0$ is a member of the sequence.