A is a set formed by the limit/accumulation points of the B set. What are the elements of A?

Question

The set B looks like this.

$ B=\{\sin(\frac{\pi n}{4}+\frac{1}{n}),n \in \mathbb{N} \}$

How can I determine the accumulation points?

Answer 1

If $n$ is a multiple of $4$ then $\sin(\pi n /4+1/n) = (-1)^{n/4}\sin(1/n)$ whose limit is $0$ when $n\to\infty$.
You can find the remaining by letting $n=4m+r$ with different integer values for $r$.