Finiteness of an algebraic variety $V(I) \subseteq \mathbb{C}^n$ where $I$ is a zero dimensional ideal of $\mathbb{C}[x_1,\dots,x_n]$

Question

How to demonstrate this?:

Let I be an ideal of $\mathbb{C}[x_1,\dots,x_n]$ such that $\frac{\mathbb{C}[x_1,\dots,x_n]}{I}$ has finite dimension ($I$ is a zero dimensional ideal). Then $V(I) \subseteq \mathbb{C}^n$ is finite.

I have thought about using a theorem stating that $I$ is a zero dimensional ideal if and only if for each $i$ exists $g_i \in G$ ($G$ is a Gröbner basis of $I$) and $\beta_i \in \mathbb{N}$ such that LM($g_i$)=$x_i^{\beta_i}$ (LM refers to leading monomial).

Answer 1

Indeed, this is the computational way to show that an ideal is zero-dimensional. I'd consider a computer algebra system such as Singular to compute a reduced Gröbner basis. Do you have a concrete example?