I am dealing with the following problem.
Let $X$ be an affine variety of positive dimension. For a point $x \in X$, there exists a nonzero function $f \in A(X)$, where $A(X)$ is the coordinate ring of $X$, such that $f(x) = 0$.
I feel this claim is not too hard but I cannot find a proper way to prove it. One idea that occurs to me is that a point in $X$ has a one-to-one correspondence to a maximal ideal of $A(X)$, i.e. generating by functions vanishing at this point. So we only need to prove that the maximal ideal contains an element other than $0$, which seems true by weak Nullstellensatz. But where does the condition "of positive dimension" goes? Or the thing is worse that my idea is completely wrong. Hopefully someone could give me some advice.