Lemma 2.2.2. Put \(D = [0, 1) \times \prod_p \mathbb{Z}_p \subset \mathbb{A}\). Every \(x \in \mathbb{A}\) can be written in a unique way as \begin{equation} \tag{2.2.2.1} x = \gamma + d, \qquad \gamma \in \mathbb{Q},\ d \in D. \end{equation}