Proposition 1

Let $(E_\alpha, f_{\alpha\beta}),\ (E'_\alpha, f'_{\alpha\beta}),\ (E''_\alpha, f''_{\alpha\beta})$ be three inverse systems of $A_\alpha$-modules and $(u_\alpha),\ (v_\alpha)$ two inverse systems of $A_\alpha$-linear mappings such that the sequences

are exact for all $\alpha$. Then, writing $u = \lim u_\alpha,\ v = \lim v_\alpha$, the sequence

is exact.

As $u_\alpha^{-1}(0) = \{0\}$ for all $\alpha$, it follows from Set Theory, III, § 7, no. 2, Proposition 2 that $u^{-1}(0) = \{0\}$, hence $u$ is injective; further, the $u_\alpha(E'_\alpha)$ form an inverse system of subsets of the $E_\alpha$ and thus $u(\lim E'_\alpha) = \lim u_\alpha(E'_\alpha)$. As $u_\alpha(E'_\alpha) = v_\alpha^{-1}(0)$ by hypothesis, $v^{-1}(0) = \lim u_\alpha(E'_\alpha) = u(\lim E'_\alpha)$ (Set Theory, III, § 7, no. 2, Proposition 2), which completes the proof.