Let us assume that a group $G$ has a filtration $\{N_i\}_{i\in \mathbb{N}}$. Let $A$ be a normal subgroup of $G$. I want to know if it is always $$\bigcap_{i\in \mathbb{N}}N_iA=A?$$
I have not been able to find a counterexample or nor a proof. It seems to me that it should be true but I am not sure how to prove it.
Any help will be appreciated.