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Let $\tilde{F}_m$ be the algebra of germs at $m$, and $F_m\subset \tilde{F}_m$ be the ideal of germs where $\forall {\mathbf{f}}\in F_m$ we have that ${\mathbf{f}}(m)=0$.

I am not understanding immediately why $$\tilde{F}_m\supset F_m\supset F_m^2\supset F_m^3\supset\cdots$$ I am only confused about situations such as $f\in F_m$ then $f^3\in F_m^3$ but why is $f^3\in F_m^2$? This should only be possible if $f^{1.5}\in F_m$, but what would that even mean? Why would this function exist?

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    In this case your question happens to be answerable as is, but in general you should try to provide more context for a question like this. For instance, what kind of germs are you talking about? What is $m$? – 2017-01-01

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If $I$ is an ideal in a ring, then $I^n$ denotes the ideal generated by all products of the form $i_1\cdots i_n$ for $i_1,\dots,i_n\in I$, not just the set of elements of the form $i^n$ for $i\in I$. So for instance, if $f\in I$ then $f^3\in I^2$ since $f^3=f^2\cdot f$ and $f,f^2\in I$. More generally, $I^n\subseteq I^{n-1}$ since you can write $i_1\cdots i_n$ as $i_1\cdots (i_{n-1} i_n)$ and $i_{n-1}i_n\in I$.

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    Thank you, that was silly of me! (Also, I agree with your comment on the question, will do) – 2017-01-01