*Context* (not necessary to understand the task). I'm working through so called Eigenvalue Estimation Problem from the field of Quantum Computing. More precisely, I try to measure the inaccuracy of the quantum circuit when an eigenvalue measured is irrational (the circuit is designed for rational phases).

*The task* can be reduced to the estimation with the lower bound of an (modulus of) expression:

$$
\frac 1 {2^m} \frac {1 - e^{2\pi i \delta 2^m}} {1 - e^{2\pi i \delta }}
$$

Here $m$ is a natural number, $\delta$ is a real from $[0,1)$, $i$ is the imaginary unit.

It is known that the desired lower bound is $2 / \pi$. It is suggested to get upper bound for denominator (which is clearly $2\pi\delta$) and lower bound for numerator (which is problematic for me) and then divide them. It is known that numerator $N$ can be estimated like this: $ N > 2\pi\delta2^m(2/\pi) > 4\delta 2^m$. Can anyone explain, how to obtain the bound in the left part of the inequality?