  Let $\mathcal{A}_n$ be the $2$-part of the ideal class group of the $n$-th
layer of the cyclotomic $\mathbb{Z}_2$-extension of a real quadratic number
field $F$. The cardinality of $\mathcal{A}_n$ is related to the index of
cyclotomic units in the full group of units. We present a method to study the
latter index. As an application we show that the sequence of the
$\mathcal{A}_n$'s stabilizes for the real fields $F=\mathbb{Q}(\sqrt{f})$ for
any integer $0<f<10000$. Equivalently Greenberg's conjecture holds for those
fields.
