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Let $(R,m)$ be local, $\dim R= \operatorname{depth}R=d$. Prove that every system of parameters is an $R$-sequence.

Here is my draft: $grade(I)=grade(\sqrt{I})=grade(m)=ht(m)=d \le k$ with $I=\left, (a_{i})$ is a system of parameters.

However, that is all I have. I can not prove $k\le d$. Can you help me check it?

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    What is $\operatorname{gr}$? – 2017-02-20
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    It is grade.... – 2017-02-20
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    You should define it. I guess it is just the depth of $R$? – 2017-02-20
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    you mean $depth R= dim R/Ann(R)$ right? I add the definition to my post ? – 2017-02-20
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    For the case $d=1$ you have to prove: If there is a non-zero divisor in $\mathfrak m$, then any zero divisor cannot form a system of parameters. This is very easy. The case $d>1$ is a trivial induction. You just need that both the dimension of $R$ and the depth of $R$ drop by $1$ if you divide out a non-zero divisor. – 2017-02-20
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    Thank you so much. My stupid mistake – 2017-02-20
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    Soulostar, please add the missing definitions to the question. We probably want to remove the comments, but before we do that you should improve the question. – 2017-04-09

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