Invertible plus nilpotent implies invertible

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.