Ground state energy

I am trying to get familiar with the ground state energy of an operator. In my lecture we defined the ground state energy of a self-adjoint operator $H$ that is bounded from below as
$$ E_0= inf_{psiin D(H)setminus{0}} frac{langle psi, Hpsirangle}{||psi||^2}= inf sigma(H)$$

What I quite dont understand is, why second equality. Why is this the same as the spectrum of the operator? Does it have to do with the spectral theorem? I tried to show the equality in the following way:

$“geq”$
begin{align}
langle psi, Hpsirangle&=int_{sigma(H)}lambda ~text{d}mu_{psi,psi}
&geq inf_{lambdain sigma(H)} lambda int_{sigma(H)} text{d}mu_{psi,psi}
&= left( inf sigma(H)right)||psi||^2
Rightarrow E_0 geq inf sigma(H)
end{align}

$“leq”$ Let $varepsilon>0$. Find $psiin D(H)$ with $||psi||=1$, such that
begin{align}
langle psi, Hpsirangle&leq infsigma(H)+varepsilon
end{align}
Since $varepsilon$ was chosen arbitrarily, we find that $E_0leq inf sigma(H)$.

Does this make any sense?

I also have to show that for an essentially self-adjoint operator $(H,D(H))$ and its self-adjoint closure $(bar{H},D(bar{H}))$ the ground state energy $E_0$ of $H$ agrees with $bar{E_0}$ of $bar{H}$. I don’t really know why this statement is true, maybe because I did not understand the concept of ground state energy yet…
I would be really grateful for any help!