Doubt Using the Alternating Series Test

I want to prove the following equality

$$1-\frac{x^2}{6}<\frac{\sin(x)}{x}<1$$ using the alternating series test

To do this, first consider what the proof says, that is, that the series $\sum_{n=k}^\infty a_n$ can be expressed as $a_n=(-1)^n b_n$ and if $\lim_{n\to \infty} b_n=0$ and b_n is decreasing then $\sum_{n=k}^\infty a_n$ converges.

I think, that taking the fact that $b_n$ is decreasing can help me to start a work with a timely inequality, however what would be the choice of $b_n$ and $a_n$, I really don't have very clear how to use the proof, any suggestions? any help is appreciated!