Why do the k-fold products of the germs of functions that vanish on a neighborhood of a point m form a descending sequence of ideals?

Question

I'm going through Warner's Foundations of Differentiable Manifolds and Lie Groups, and he loves the idea of defining the tangent vectors at $p \in M$ using the germs of functions that agree on a neighborhood of $p$, which he denotes $\tilde{F}_p$.

He writes that the powers of the ideal formed by the germs which vanish in a neighborhood of $p$, denoted $F_p^k$ and representing the ideal formed by the k-fold products of elements in $F_p$, form a descending sequence of ideals:

$$ \tilde{F}_p \supset F_p \supset F_p^2 \supset F_p^3 \supset \dots $$

It's unclear to me why this claim is true. I need to understand this to make sense of the quotient he proposes afterward, namely $F_p / F_p^2$.

Could someone clarify?

Answer 1

After some reflection and discussion, it is now clear that Warner's statement is true in the sense that these spaces, $F_p^{j+1}$, represent those vanishing germs whose $j^{th}$ derivative vanishes at the point $p$. In that sense, it's clear that if the $(j-1)^{th}$ derivative vanishes, so does the $j^{th}$.

Answer 2

If a germ belongs to $F_p^2$ then it is the product of two elements of $F_p$, so it still vanishes but up to "order 2". $F_p/F^2_p$ is just the germs that vanish modulo the germs which vanish "twice". So you get "differentials".