In the Wikipedia article on Absolute value (algebra), the completion of an integral domain is defined as the quotient ring of Cauchy sequences by null sequences. The integral domain is then embedded in this quotient ring.
Considering the integral domain of integers, can someone provide a simple example of such an embedding ?
The embedding you propose is fine. In all treatments of completion I have seen, one first proves that the map sending elements of the integral domain to constant sequences modulo null-sequences is preserving absolute values, where the absolute value on the completion is defined by
$ |[(a_k)]|:=\lim\limits_{k\rightarrow\infty}|a_k|, $
where $[(a_k)]$ denotes the equivalence class of the Cauchy sequence $(a_k)$. Injectivity then follows. Of course to prove norm preservation one has to work a bit ...