Algebra of bounded polynomials on a set Zariski closed at infinity cannot be finitely generated

We prove that if a set S⊂Rⁿ$S\subset {\mathbb{R}}^n$ is Zariski closed at infinity, then the algebra of polynomials bounded on S cannot be finitely generated. It is a new proof of a fact already known to Plaumann and Scheiderer (2012) 1]. On the way we show that if the ring R_ζ1,…,_ζk]⊂RX]$\mathbb{R}{\zeta }_1,\dots ,{\zeta }_k]\subset \mathbb{R}X]$ contains the ideal (_ζ1,…,_ζk)RX]$({\zeta }_1,\dots ,{\zeta }_k)\mathbb{R}X]$, then the mapping (_ζ1,…,_ζk):Rⁿ→R^k$({\zeta }_1,\dots ,{\zeta }_k):{\mathbb{R}}^n\to {\mathbb{R}}^k$ is finite.