I'm trying to understand the definition of "direct limit" in Exercise 2.14 of Introduction to Commutative Algebra, M. F. Atiyah & I. G. MacDonald:
A partially ordered set $I$ is said to be a directed set if for each pair $i,j$ in $I$ there exists $k\in I$ such that $i\leq k$ and $j\leq k$.
Let $A$ be a ring (rmk. all rings in this book are commutative with indentity $1$), let $I$ be a directed set and let $(M_i)_{i\in I}$ be a family of $A$-modules indexed by $I$. For each pair $i,j$ in $I$ such that $i\leq j$, let $\mu_{ij}\colon M_i\to M_j$ be an $A$-homomorphism, and suppose that $\mu_{ii} = \text{id}_{M_i}$ and $\mu_{ik} = \mu_{jk}\circ\mu_{ij}$ whenever $i\leq j\leq k$. Then the modules $M_i$ and homomorphisms $\mu_{ij}$ are said to form a direct system $\mathbf{M} = (M_i, \mu_{ij})$ over the directed set $I$.
We shall construct an $A$-module $M$ called the direct limit of the direct system $\mathbf{M}$. Let $C$ be the direct sum of the $M_i$, and identify each module $M_i$ with its canonical image in $C$. Let $D$ be the submodule of $C$ generated by all elements of the form $x_i-\mu_{ij}(x_i)$ where $i\leq j$ and $x_i\in M_i$. Let $M = C/D$, let $\mu\colon C\to M$ be the projection and let $\mu_i$ be the restriction of $\mu$ to $M_i$.
The module $M$, or more correctly the pair consisting of $M$ and the family of homomorphisms $\mu_i\colon M_i\to M$, is called the direct limit of the direct system $\mathbf{M}$, and is written $\displaystyle\lim_{\to}\mathbf{M}$.
Here are my questions:
(1) How is this "limit" related to other easy limits, e.g., the limit of a sequence?
(2) Why should $D = \{(x_i - \mu_{ij}(x_i)\}$ be killed in $C$?
MY ATTEMPTS
The properties $\mu_{ii} = \text{id}$ and $\mu_{ik} = \mu_{jk}\circ\mu_{ij}$ seem to imply connectedness of the direct system. If we have $x_i\in M_i\hookrightarrow \mathbf{M}$, then it is equivalent to $\mu_{ij}(x_i)$ for all $j\geq i$, and $j$ can be larger ones as $I$ is a directed set. This may imply convergence and limits.
