Sequence Converge or Diverge

Question

Does the following sequence converge or diverge: $a_n=\frac{\sin{\left(n\right)}}{2^{n}}$? My initial thought was that any value of $n$ to $\sin$ will be less than $1$.

Answer 1

Hint: $|\frac{\sin(n)}{2^n}|\le \frac{1}{2^n}$

Answer 2

Your intuition is right.

In general:

If $x_n$ is a bounded sequence and $y_n \to 0$, then $x_n y_n \to 0$.

Here is a proof:

If $|x_n| < M, |y_n-0| < \varepsilon/M$ for $n$ sufficiently large, then $|x_n y_n-0| < \varepsilon$.