Conditions for the Hamiltonian's spectrum to be discrete

I came across this article [1], in which the author studies some Hamiltonian that have a discrete spectrum even though they do not go to infinity at infinity.

In there, the author makes several claims, that I don’t really get :

If $H_1 geqslant H_2$ and $H_2$ has a discrete spectrum, then so does $H_1$.

If $operatorname{Tr}e^{-tH}< infty$ for any $t$, then $H$ has a discrete spectrum.

I would like to understand the intuition behind those results, as well as their formal proofs.

[1] Some quantum operators with discrete spectrum but classically continuous spectrum; B. Simon (Caltech); Published in: Annals Phys. 146 (1983), 209-220