Existence of subgraphs with vertices of large degree [Diestel's book] 
This is an excerpt from Diestel's book (fifth edition).  Since the graph $G=(V,E)$ has at least one edge, i.e. $|E|\geq1$ then $\varepsilon(G)=\dfrac{|E|}{|V|}>0.$ How it follows that none of the graphs in our sequence is trivial?
Here is my approach: Since $\varepsilon(G_{i+1})\geq \varepsilon(G_{i})$ then $\varepsilon (G_i)\geq \varepsilon(G)>0$. This sounds a bit stupid: But if $\varepsilon(G_i)>0$ how it implies that $G_i\neq \varnothing$?
I am totally confused.

The point is that you only remove one vertex at each step. So the only way to reach $G_i=\varnothing$ is if $G_{i-1}=K_1$, but that would mean $\epsilon(G_{i-1})=0$ which is not true.
