Let $K$ be a field and $P$ a homogeneous prime ideal in $K[x_1,\ldots,x_n]$, with height $r$. I want to show that there is a chain of homogeneous primes $P_0\subsetneqq P_1\subsetneqq\cdots\subsetneqq P_r$ such that $P_r = P$.
First I tried to use this theorem:
Let $R$ be a Noetherian ring and $P$ an ideal prime of height $r$, then there are $a_1,\ldots,a_r\in R$ such that $P$ is minimal over $(a_1,\ldots,a_r)$.
But this result couldn't help me. After that I tried to use some results about transcendence degree, and I couldn't make any progress. I need some help to see what to do here. Any help is welcome, thank you!