In my textbook, when proving that
$${x_n} = \frac{\sin1}{2} + \frac{\sin2}{2^2} + \dots + \frac{\sin n}{2^n}$$
is convergent with help of Cauchy's convergence test,
it's required that
$$0<\epsilon<1$$
on the following step of the proof:
$$\vert {x_{n+p} - x_n \vert}<\frac{1}{2^n}<\epsilon \Rightarrow 2^n>\frac{1}{\epsilon} \Rightarrow n\ln2 > -\ln\epsilon \Rightarrow n>-\frac{\ln\epsilon}{\ln2}$$
Why do we need epsilon to be smaller than 1 and greater than 0?
Thank you a lot.