Here is the notation & motivation.:
Conjecture: If $f \in \hat{m}$, then $f = \sum g_i f_i$ where each $g_i$ is homogeneous.
I've tried proof by induction on degree of $f$, number of generators making up $f$, breaking up the $f_i$ into degree buckets. What am I not seeing?
From the other post. Let $I = $ the ideal generated by all homogeneous elements. $I = \bigoplus_{n\geq 0} S_n \cap I$. So working componentwise, $h_i \in S_n \cap I = \{ \sum g_i f_i : g_i \in S, \deg(g_i f_i) = n\}$. If $g_i$ were not homogenous, then $g_i f_i = h_{\deg = n} + h_{\deg \neq n}$ and we can move the $h_{\neq n}$ part to another homogeneous part, inductively reducing the degree if necessary.
That is too informal. $S$ is graded usually by $S_n = \{ $ polynomials of degree $n \}$ that is with possible lower degree terms. Now grade $S$ by $S'_n = \{$ homogeneous polynomials of degree $n \}$. We have $S_i S_j = S_{i + j}$ so that this is a grading, and each $S_i$ is a $k$-module.
Now $I \cap S'_n = \{ $ homogeneous polynomials of degree $n \} = S'_n$. Thus, each generator $f_i$ is in some $S'_{n_i}$ and we can write $f = \sum g_i f_i$ where the generators $f_i$ can occur more than once.
So I must have misread what the book was saying.