A Bernstein--Walsh Type Inequality and Applications A Bernstein--Walsh type inequality for $C^{\infty }$ functions of several variables is derived, which then is applied to obtain analogs and generalizations of the following classical theorems: (1) Bochnak--Siciak theorem: a $C^{\infty }$\ function on $\mathbb{R}^{n}$ that is real analytic on every line is real analytic; (2) Zorn--Lelong theorem: if a double power series $F(x,y)$\ converges on a set of lines of positive capacity then $F(x,y)$\ is convergent; (3) Abhyankar--Moh--Sathaye theorem: the transfinite diameter of the convergence set of a divergent series is zero. Keywords:Bernstein-Walsh inequality, convergence sets, analytic functions, ultradifferentiable functions, formal power seriesCategories:32A05, 26E05