Given a descending semi-exact sequence $C$ of $R-$modules
$$ C: \cdot \cdot \cdot \to C_{n+1} \to C_{n} \to C_{n-1} \to \cdot \cdot \cdot $$, consider the direct sum $$ X= \sum_{n \in \mathbb{Z}} C_n $$ and the restriction $ d: X \to X $ of the cartesian product of all edge operators $ \partial_n: C_{n} \to C_{n-1} $. I need to prove that $ d \circ d=0$ and $H(X)= \displaystyle \sum_{n \in \mathbb{Z}} H_n(C)$. Where $H(X)= Ker(d) / Im(d)$ and $H_n (C)= Ker ( \partial_{n}) / Im (\partial_{n+1})$