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How can I use prime avoidance to prove that if $R$ is a Noetherian ring and $J\subset R$ an ideal with $\mathrm{height}(J)=n$ then there exists $x_1,\ldots,x_n\in J$ such that $\mathrm{height}((x_1,\ldots,x_i))=i$ for all $i=1,\ldots,n$?

I know that if $R$ is local and Noetherian of dimension $d$ then there exist $x_1,\ldots,x_d$ such that $\mathrm{height}(x_1,\ldots,x_i)\geq i$ for all $i=1\ldots,d$.

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