The following property of ultrametric spaces seems quite strange:
(No new values of the metric after completion) Let $x_1, x_2, \ldots$ be a sequence in $X$ converging to $x \in X$. Suppose $a \in X, a \ne x$. Then $d(x_n, a) = d(x,a)$ for large $n$. [...] The metric completion of $X$ is again an ultrametric space, denoted by $X'$, with the metric on $X'$ also denoted by $d$. By the above remark we have $d(X\times X) = d(X' \times X')$.
What does this mean, "no new values after completion" could be interpreted that the space equals it's own completion, but that I am sure is not the case. So what does this property mean, and does it has any applications?
Remark: This is taken from page 4 of this book.