Let $\varepsilon > 0$. Since $x$ is Cauchy, there exists $N \in \mathbb{N}$ such that $d(x_n, x_m) < \varepsilon$ for every $n, m \geq N$. Thus, if $n \geq N$, then
\begin{equation}
d(x_0, x_n)
\leq d(x_0, x_N) + d(x_N, x_n)
< d(x_0, x_N) + \varepsilon
\end{equation}
The set $\{ d(x_0, x_n) : 0 \leq n \leq N \}$ is finite and nonempty, so result
R1268: Non-empty finite subsets of classical number sets contain extended extrema guarantees that it contains a maximum, say $K_{\text{max}}$. Setting $K : = d(x_0, x_N) + \varepsilon$ and choosing
\begin{equation}
r
: = \max(K, K_{\text{max}}) + 1
\end{equation}
and $a : = x_0$, then we have $d(a, x_n) < r$ for every $n \in \mathbb{N}$ and the claim follows. $\square$
