Quotienting out nilpotent elements

Let $R$ be a commutative ring with $1$ , and $N$ the ideal

N = {a \in R ∣ a^{n} = 0 for some n} .

Let $[b]$ be the image of $b \in R$ in $R / N$ . Prove that if $[a] \in R / N$ and $[a]^{m} = 0$ then $[a] = [0]$ .

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Let $R$ be a commutative ring with $1$, and $N$ the ideal
\[
	N=\{a\in R\,\mid\, a^n=0\text{ for some }n\}.
\]
Let $[b]$ be the image of $b\in R$ in $R/N$. Prove that if $[a]\in R/N$ and $[a]^m=0$ then $[a]=[0]$.