I read the Brezis functional analysis, differential equations and Sobolev spaces textbook. We extend $g(x):G\to\mathbb{R}$ to $f(x):E\to\mathbb{R}$
The idea of the proof is to construct a well-ordered set of extension functions $g\leq h_1\leq h_2\leq\dots\leq f$ which domains are embedded $G\subset\dots\subset E$, and then by Zorn lemma have a maximal element of such set. It is rest to prove that the domain of the maximal element $Dom(f)$ is indeed equal to $E$. Further it goes by contradiction and we consider $x_0\notin Dom(f)$, but then we set $$h(x+tx_0)=f(x)+\alpha t$$ and I don't understand how does one naturally come up with that setting