Equivalence of Definitions of Reachable

Proof
Let $G = \struct {V, A}$ be a directed graph.

Let $u, v \in V$.

Let $\RR$ be the reachability relation for $G$, defined as the transitive closure of $A$.

Definition 1 implies Definition 2
Suppose that $v$ is reachable from $u$ by definition 1.

Then there exists a directed walk $\tuple {u = x_0, \dots, x_n = v}$ from $u$ to $v$.

Then by the definition of directed walk:
 * $x_0 \mathrel \RR x_1 \mathrel \RR \cdots \mathrel \RR x_n$

Then by the definition of transitive closure:
 * $u \mathrel \RR v$

Thus $v$ is reachable from $u$ by definition 2.

Definition 2 implies Definition 1
Suppose that $v$ is reachable from $u$ by definition 2.

That is:
 * $u \mathrel \RR v$

Thus by the definition of transitive closure:
 * $\exists x_0, \ldots, x_n \in V: x_0 \mathrel \RR x_1 \mathrel \RR \cdots \mathrel \RR x_n$

Then $\tuple {x_0 \mathrel \RR x_1 \mathrel \RR \cdots \mathrel \RR x_n}$ is a directed walk from $u$ to $v$.

Therefore, $v$ is reachable from $u$ by definition 1.