claim.

Proof of Theorem \[inv-2\] {#cheap}
==========================

We now prove Theorem \[inv-2\]. It will be convenient to reformulate the result in terms of a “cheap” form of nonstandard analysis (as used in [@cheap]), involving sequences of potential counterexamples. The full machinery of nonstandard analysis, such as ultraproducts and the construction of Loeb measure, will not be needed for this reformulation; one could certainly insert such machinery into the arguments below, but they do not appear to dramatically simplify the proofs.

We will need a natural number parameter $n$. In the sequel, all mathematical objects will be permitted to depend on this parameter (and can thus be viewed as a sequence of objects), unless explicitly declared to be “fixed”. Usually we will suppress the dependence on $n$. For instance, a sequence $G_n$ of compact abelian groups will be abbreviated as $G = G_n$. A real number $x = x_n$ depending on $n$ is said to be *infinitesimal* if one has $\lim_{n \to \infty} x_n = 0$, in which case we write $x = o(1)$. If $x = x_n$, $y = y_n$ are real numbers such that $|x_n| \leq Cy_n$ for all sufficiently large $n$ and some fixed $C>0$, we write $x 