Let $\text{codim}(E_i)=1$ for $i=1,...n$. Prove that $\text{codim}(\cap E_i) \le n$.
I was trying to prove it only with simple dimensions calculation - codim$(\cap E_i) = \dim(E\setminus \cap E_i) = \dim(E\setminus E_1 \cap \dots \cap E_n) \le \dim(E\setminus E_1) + \dots + \dim(E\setminus E_i) = \text{codim}(E_1)+\dots + \text{codim}(E_i) =n $
But are all my equations and inequalities correct?
If not, any other ideas?