This is a result claimed in Example 6.17 of All of Statistics.
Basically, we need to prove that:
$$ z_{\alpha/2} \sqrt{\frac{\hat p_n (1- \hat p_n)}{n}} < \sqrt{\frac 1 {2n} \log(\frac 2 \alpha)} $$
We can use the inequality $p(1-p) \le \frac 1 4$ and make the substitution that $t=\alpha/2$. We only need to prove that $$ \sqrt{-2\log t} > z_t $$
The approach I used requires quite a lot of work:
Since the CDF of the standard normal distribution $\Phi$ is monotonic, it suffices to prove that (apply $\Phi$ on both sides):
$$ \int_0^{\sqrt{-2\log t}} \frac 1 {2\pi}\exp(-\frac{x^2}2)\,\mathrm dx > \frac 1 2 -t $$
Then let $g(t)=\text{LHS}\, - \, \text{RHS}$ and take its derivative. We find that g(t) first increase then decrease. Thus we only need to verify the values it takes at boundary points. After some analysis, we can show that this is always positive.
Is there a better way to do this? I've tried the Mill's inequality as well to get bounds of the tail of the normal distribution, but it only works in an interval and you still need to justify the inequality on the interval leftover.