In an article I've read in a proof that distinguishes two cases something like: "the second case can be shown by an easy coupling argument using the first case."
What is a coupling argument?
Edit Here is the context. Unfortunately this all corresponds to lot of text before, but I try to reduce it to the main things.
Lemma
If $\lim_{L\to\infty}\delta_L=0$ and $\lim_{L\to\infty}\delta_LL^{d/2}=\infty$, then $\lim_{L\to\infty}\text{Prob}_L(R)=1$.
Proof. Applying Lemma 9, it suffices to show that if $\lim_{L\to\infty}\delta_L=0$ and $\lim_{L\to\infty}\delta_L L^{d/2}=\infty$, then $\lim_{L\to\infty}\text{Prob}_L(G)=1$.
If $\lim_{L\to\infty}\delta_L^3L^d=0$, then this immediate from Lemma 7.
If the $\delta_L$'s are larger such that $\limsup_{L\to\infty}\delta_L^3L^d>0$, it is even easuer to have an adjacent 12 pair and $\lim_{L\to\infty}\text{Prob}_L(G)=1$ is proved in this case by a simple coupling argument using the first case.
Hm, here is so much notation I would have to explain that it would become a endless question. Maybe you can help neverhtless? I am not sure, sorry.