Let $(G_i,x_i)$ be a sequence of rooted graphs that we can assume to have uniformly bounded (finite) maximum degrees, and let $P_i$ be a sequence of first order formulae (in the language of graph theory) that for simplicity only have one free variable. Suppose now that the $P_i$ are such that they can be checked locally and uniformly so, that is, for each formula $P_i$ there is a natural number $R_i$ such that for any $y\in G_j$, $P_i(y)$ holds in $G_j$ if and only if it holds on $B_{G_j}(y,R_i)$.
It is then quite clear (if I am not terribly mistaken) that if there is a sequence $S_i$ such that $P_i$ holds on $B_{G_j}(x,S_i)$ for all $j\geq i$, then every point in the rooted graph $(G,x)$ that is the limit of the sequence $(G_i,x_i)$ satisfies all the formulas $P_i$.
The problem is: this behaviour arises quite often, and it obviously becomes tedious for the reader to read what is essentially the same proof over and over, so I have searched the internet for some kind of result similar to this one that I can reference to shorten my proofs, with no success. But I think that this has to have been studied before, because the same behaviour is manifested by a lot of different objects e.g. limits of functions.
Question: Is there indeed a known result that generalizes what is described in the first two paragraphs? Is there some branch of mathematics (perhaps in model theory) that studies topics related to this question?
Thank you in advance. Please, if the question needs to be made more specific or if some of the terminology is wrong tell me so I can fix it.
