= Complement of a finite union of finite-index subgroups
In an <infinite group> $G$, the complement of a finite union $H_1\cup\cdots\cup H_k$ of <finite-index subgroups> is either empty or infinite. To prove this, their intersection $L$ has finite <subgroup index> by the <finite intersection of finite-index subgroups> bound, so it is infinite: otherwise finitely many finite <left cosets> would cover $G$. If $g$ lies outside the union, the whole infinite <left coset> $gL$ does too. Indeed, $g\ell\in H_i$ and $\ell\in L\subseteq H_i$ would imply $g\in H_i$.
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