Applying the definition of convergence for a sequence

Question

There are many proofs that use $N = \max\{\dots , \dots \}$ for the choice of $N$. For example if we want to prove $a_n = \frac{\sin 1}{2} + \frac{\sin 2}{2^2} + \dots \frac{\sin n}{2^n}$ is a Cauchy sequence, we let $N = \max\{\lfloor\ln(1/\epsilon)/\ln2 \rfloor + 1, 1\} $. Why this happens sometimes in the proofs? Is it always necessary? If not, when we should choose maximum for $N$? I think it has something to do with different cases for $\epsilon$.