Troubles verifying $langle omega rangle = frac{1}{2}epsilon epsilon_0 |vec{E}_0|^2$

I’m trying to prove that the time average of the energy density $omega$ is equivalent to

$$langle omega rangle = frac{1}{2}epsilon epsilon_0 |vec{E}_0|^2$$

for a plane EMW. My approach is to use that

$$langle f cdot g rangle = frac{1}{2} Re(f cdot g^*)$$

This comes from Griffiths’ Introduction to Electrodynamics:

and also $$omega = frac{1}{2} ( epsilon epsilon_0 E^2 + mumu_0 H^2)$$

1. The time average distributes linearly so using $langle f cdot g rangle $ with $f=g=E$ yields $$langle E^2rangle = frac{1}{2}left(frac{E_0^2}{1}right)$$

2. Similarly for $H$ I get $$langle H^2 rangle = frac{1}{2}left(frac{E_0^2}{c^2}right)$$ using the fact that $vec{H} =frac{1}{c} (hat{k} times vec{E_0}) e^{i(kx-wt)}$.

3. When I sum this two quantities I get $$omega = frac{1}{2} E_0^2 left(frac{epsilonepsilon_0}{2} + frac{mumu_0}{2c^2}right)$$ but this puzzles me. I know it shouldn’t be this way; otherwise I would get $mu_0^2 = mu/epsilon$ and that’s not true from what I know.