Invertible plus nilpotent implies invertible

This article is about the statement of a simple but indispensable lemma in commutative algebra
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Statement

In a commutative unital ring, the sum of an invertible element and a nilpotent element is an invertible element.

Facts used

$1+x^{n}=(1+x)(1-x+x^{2}-\ldots +(-1)^{n-1}x^{n-1})$

Proof

Given: A commutative unital ring $A$, elements $a,x\in A$ such that $a$ is invertible, and $x^{n}=0$ for some positive integer $n$

To prove: $a+x$ is invertible

Proof: Since $a$ is invertible, it suffices to prove that $1+x/a$ is invertible. Since $x^{n}=0$, we also have $(x/a)^{n}=0$. The above formula then tells us that $1+x/a$ is invertible, completing the proof.