How does one generally show syntactical statements in PA about primitive recursive functions or relations?


For example something like: 


Let $A$ be a prim. rec. relation such that $n\in A$ for every numeral $n$. Show $PA \vdash \forall x A(x)$ (where $A$ denotes the relation as well as the formula representing the relation in PA).



Syntactical statements in PA about primitive recursive functions
Sometimes you can't prove what you asked for. For example, it is routine to give a primitive recursive definition of the predicate $A(x)$ formalizing in a natural way "$x$ is not the Gödel number of a proof in PA of a contradiction." Then, for each natural number $n$, this predicate holds of $n$ (because PA is consistent). But PA cannot prove $\forall x\,A(x)$, because, by Gödel's second incompleteness theorem, PA cannot prove its own consistency. 


