Homework Statement

On Wikipedia there is an article about perturbation theory. To understand something I need to understand the following relation. They say:

Homework Equations

$H |n> = E_n |n>$

So:

$<n| H = <n| E_n$

H is Hermitian.

So: Why is this?

The Attempt at a Solution

I don't know if this is valid.

$(H \cdot |n>)^\dagger = (E_n \cdot |n>)^\dagger$

$\Leftrightarrow$

$<n| H^\dagger = <n| E_n^\dagger$

And since H is Hermitian and E_n is real, I get:

$<n| H = <n| E_n$

Is this how things work?