Stone–Weierstrass approximation theorem for multivariable case.

Question

Stone–Weierstrass approximation theorem says "Let $A$ be a (complex) unital sub-algebra of $C(K, \mathbb{C})$, such that if $ f\in A$, then $\overline{f} \in A$, and $A$ separates points of $K$. Then $A$ is dense in $C(K, \mathbb{C})$. Here $K$ is a compact metric space." I wish to apply it for a continuous function $f$ defined on $S^n$ where $S$ is the unit circle in complex plane. In this case I consider $A$ as the collection of all trigonometric polynomials of the form $$\sum_{-k_j \le i_j \le k_j \atop{1\le j \le n}}a_{i_1,\ldots,i_n}z_1^{i_1}\ldots z_n^{i_n}; \ \ (z_1, \ldots,z_n) \in \mathbb{T}^n.$$ In this case $A$ satisfies all conditions mentioned in above approximation theorem. Thus $f$ can be approximated uniformly by a sequence of trigonometric polynomials (in $n$ variables). Am I arguing it right or something is missing?