Here is the last problem of my final exam in "Commutative algebra" which I think, no one has solved it completely, today!
Let $R$ be a (commutative with $1$) Noetherian ring. Let $a_1,...,a_n$ and $b_1,...,b_n$ be two regular sequences in $R.$ Prove that there is a regular sequence $c_1,...,c_n$ s.t. for each $i, \; 1 \leq i \leq n, \; c_i \in (a_1,...,a_i) \cap (b_1,..,b_i).$
Note: I attempted to show that $c_i=a_ib_i$ is the desired one, but it seems that we can not do anything, when $i \geq 2.$