How familiar are you with the Jacobson radical $J(R)$? It is the intersection of all the maximal ideals of a ring $R$ with 1. If $r\in J(R)$ we have $1-r$ cannot lie in any maximal ideal and so must be invertible.
To answer your question, you must show what was actually, for $M$ an ideal of $R$, in Jacobson's thesis years before it became known as Nakayama'a Lemma (with Nakayama denying he (Nakayama) was the originator).
Nakayama's Lemma Let $M_R$ be a finitely generated module over any ring $R$ with identity. Then $MJ(R)\ne M$.
To show this, start with $\{a_1,\ a_2,\ \cdots\, \ a_n\}$ a set of generators for $M$ of smallest cardinality $n$. Express $a_1$ as a sum of the form $a_1=\sum_{i=1}^n a_ij_i$ where $\{j_i\}\subseteq J(R)$. Now hopefully you can complete the proof of the theorem in your posting and edit the posting to make sure the theorem is mathematically correct.