Approximation and convergence of formal CR-mappings

Let $M\subset C^N$ be a minimal real-analytic CR-submanifold and $M’\subset C^{N’}$areal-algebraicsubsetthroughpoints$p\in M$and$p’\in M’$. We show that that any formal (holomorphic) mapping $f\text{\hspace{0.17em}⁣}:(C^N,p)\to (C^{N^{\mathrm{\prime }}},p^{\mathrm{\prime }})$, sending $M$ into $M^{\mathrm{\prime }}$, can be approximated up to any given order at $p$ by a convergent map sending $M$ into $M^{\mathrm{\prime }}$. If $M$ is furthermore generic, we also show that any such map $f$, that is not convergent, must send (in an appropriate sense) $M$ into the set $E^{\mathrm{\prime }}\subset M^{\mathrm{\prime }}$ of points of D’Angelo infinite type. Therefore, if $M^{\mathrm{\prime }}$ does not contain any nontrivial complex-analytic subvariety through $p^{\mathrm{\prime }}$, any formal map $f$ as above is necessarily convergent.