In their paper (with full text here) Bethuel et al. use an argument on page 636 that (simplified) reads:
Assume you have a sequence of functions $(v^n)_{n \in \mathbb{N}}$ with domain $\mathbb{R}^N$, that are uniformly bounded in $C^k$, i.e. for every $k \in \mathbb{N}$ there is a constant $K(k)$, s.t. \begin{equation} \Vert v^n \Vert_{C^k(\mathbb{R}^N)} \leq K(k).\end{equation}
They then use Ascoli's theorem to assert that for every compact ball $B(0,j)$ there is a smooth map $\mathfrak{v}^j$ s.t. for $n \to \infty$ \begin{equation} v^n \to \mathfrak{v}^j ~~~ \mbox{ in } C^k(B(0,j)), \forall k \in \mathbb{N}.\end{equation}
This part I do understand. The next step is to show that there is a function $\mathfrak{v}$, s.t. $v^n \to \mathfrak{v}$ in $C^k(K)$ for any $K \subset \subset \mathbb{R}^N$. Their (full) argument is:
"To conclude we let $j \to \infty$, and we invoke a diagonal argument, so that in particular $\mathfrak{v} = \mathfrak{v}^j$ does not depend on the ball $B(0,j)$."
What do they mean by "diagonal argument"? Can anyone provide me with any source containing an example for such an argument?
The argument the closest to being "diagonal" that I can think of is that $v^n(x)\to\mathfrak v^j(x)$ for every $j$ and every $x$ in $B(0,j)$. By the uniqueness of the pointwise limit, for every $i$ and $j$, the functions $\mathfrak v^i:B(0,i)\to\mathbb R$ and $\mathfrak v^j:B(0,j)\to\mathbb R$ coincide on $B(0,\min(i,j))$.
One can define a function $\mathfrak v:\mathbb R^N\to\mathbb R$ unambiguously by $\mathfrak v(x)=\mathfrak v^j(x)$ for every $x$ and every $j\gt|x|$. Then $v^n\to\mathfrak v$ in $C^k(B(0,j))$, for every $j$, hence $v^n\to\mathfrak v$ in $C^k(K)$, for every bounded subset $K\subset\mathbb R^N$.