I'm working on the following problem form Nielsen and Chuang (Ex. 4.40) regarding the universality of Hadamard, phase and $\pi/8$ gates for single qubit gates:
For arbitrary $\alpha$, $\beta$,
$E(R_{\vec{n}}(\alpha), R_{\vec{n}}(\alpha + \beta)) = |1 - exp(i\beta/2)|$, where $E(U, V) = max_{|\psi\rangle} || (U-V)|\psi\rangle||$
Show (using the previous equality) $\forall \epsilon >0$ , $\exists n$, such that $E(R(\alpha)_{\vec{n}}, {R(\theta)_{\vec{n}}}^n) < \frac{\epsilon}{3}$, where $\theta$ is an irrational multiple of $2\pi$.
My attempt:
Prior to the exercise, it discusses how for any arbitrary error $\epsilon > 0$, we can find some multiple $(mod$ $2\pi)$ of irrational $\theta$ (found using the Pigeonhole principle, which I think I understand, and I'm skipping and taking as given), call it $\theta_k \in (0, 2\pi]$, |$\theta_k| < \epsilon$ such that all multiples $(mod$ $2\pi)$ of $\theta_k$ differ by at most $\epsilon$. Thus $\forall \alpha \in [0, 2\pi)$, $\exists m$, such that $|\alpha - (m\theta_k) | =|\alpha - (mk\theta)| < \epsilon$. Let $n = mk$, then
$E(R(\alpha), {R_{\vec{n}}}(\theta)^n) = E(R_{\vec{n}}(\alpha), {R_{\vec{n}}}(n\theta)) = E(R_{\vec{n}}(\alpha), {R_{\vec{n}}}(\alpha + (n\theta -\alpha))) < E(R_{\vec{n}}(\alpha), {R_{\vec{n}}}(\alpha + \epsilon)) = |1 - exp(i\epsilon/2)| = \sqrt{((1 - cos(\epsilon/2))^2 + sin(\epsilon/2)^2} = \sqrt{2(1 - cos(\epsilon/2))} < 2$
(which is not correct).
I feel like I'm missing something very obvious in this whole exercise. A hint would be appreciated.