TITLE: Is this result on the bound of regularity of an ideal true?
QUESTION [5 upvotes]: I am solving a problem in which i need to use the following result but i am not sure whether the result is true on not:

If the ideals $I_0,...,I_n$ are generated by linear polynomials in $k[x_0,...,x_n]$ then $\mathrm{reg}(I_0+\cap_{i=1}^n I_i) \leq n$.

Any counterexample or ideas to prove the above result?

REPLY [3 votes]: The result is true, and it is a particular case of Proposition 2.2 from this paper.