Let $I, p_{ij} : A_i \to A_j$ be a directed system of maps such that $p_{jk}\circ p_{ij} = p_{ik}$ whenever $i \leq j\leq k$ and $p_{ii} = \text{id}$. Directed means that for any $i,j \in I$ there exists $k \in I$ such that $i,j \leq k$. $I$ is a partially ordered set and $A_i, p_{ij}$ are general sets and maps.
Consider the maps $q_i : A_i \to \lim_{\rightarrow} A_i$ ie. into the direct limit, be defined by $q_i(a) = \bar{a} = $ the equivalence class of $a \in \lim_{\rightarrow} A_i$. If the directed system maps $p_{ij}$ are all injective then all of the inclusion maps $q_i$ are injective.
I have suppose $p_i(a) = p_j(b) = \bar{a} = \bar{b} \implies a \sim b \iff \exists i,j \leq k$ such that $p_{ik}(a) = p_{jk}(b)$. Then I'm stuck.
You want to start by supposing that $a,b\in A_i$ and $q_i(a)=q_i(b)$, i.e., that $\bar a=\bar b$, and your goal is to show that $a=b$. Since $\bar a=\bar b$, you know that there is some $k$ such that $i\le k$ and $p_{ik}(a)=p_{ik}(b)$. And therefore ... ?