Theorem 40.2:

(i) Every directed system of groups or abelian groups has an inductive limit which is unique upto isomorphism.

(ii) With the notations as in the definition (40.2), assume that $f_{\alpha}(x) = 0$ for some $x \in G_{\alpha}$. There exists $\beta \geq \alpha$ such that $f_{\alpha\beta}(x) = 0$.