  In this short note we give an answer to the following question. Let $X$ be a
locally compact metric space with group of isometries $G$. Let $\{g_i\}$ be a
net in $G$ for which $g_ix$ converges to $y$, for some $x,y\in X$. What can we
say about the convergence of $\{g_i\}$? We show that there exist a subnet
$\{g_j\}$ of $\{g_i\}$ and an isometry $f:C_x\to X$ such that $g_{j}$ converges
to $f$ pointwise on $C_x$ and $f(C_x)=C_{f(x)}$, where $C_x$ and $C_y$ denote
the pseudo-components of $x$ and $y$ respectively. Applying this we give short
proofs of the van Dantzig--van der Waerden theorem (1928) and Gao--Kechris
theorem (2003).
