Exactness of associated graded modules for induced filtrations ($0\to \mathrm{gr}N\to \mathrm{gr}M\to \mathrm{gr}(M/N)\to 0$)

Give $N$ the subspace filtration and $M/N$ the quotient filtration. If the degree-$i$ class of $m\in F_{i}M$ maps to zero, then $m=n+m^{\prime }$ with $n\in N$ and $m^{\prime }\in F_{i-1}M$. Thus $n\in N\cap F_{i}M$ and the class comes from $\mathrm{gr}N$. Inclusion is injective in every degree and the quotient map is surjective, proving the displayed short exact sequence of associated graded modules. Exactness can fail for arbitrarily chosen incompatible filtrations.