A nonstandard proof of continuity of affine varieties

Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{F}=\{f_{\lambda }:\lambda \in \mathrm{\Lambda }\}$ is a set of polynomials in $\mathbf{C}[t_1,\dots ,t_n]$ and if ${}^{\ast }\mathcal{G}=\{g_{\lambda }:\lambda \in \mathrm{\Lambda }\}$ is a set of polynomials in ${}^{\ast }{\mathbf{C}}_0[t_1,\dots ,t_n]$ such that $g_{\lambda }$ is an infinitesimal deformation of $f_{\lambda }$ for all $\lambda \in \mathrm{\Lambda }$, then the nonstandard affine variety ${}^{\ast }{V}_0(\mathcal{G})$ is an infinitesimal deformation of the affine variety $V(\mathcal{F})$.