Equivalence of Definitions of Reachable

Theorem
The following definitions of reachable are equivalent:

Proof
Let $G = (V, A)$ be a directed graph.

Let $u, v \in V$.

Let $\mathrel R$ 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 $(u = x_0, \dots, x_n = v)$ from $u$ to $v$.

Then by the definition of directed walk, $x_0 \mathrel{A} \dots \mathrel{A} x_n$.

Then by the definition of transitive closure, $u \mathrel{\mathcal R} v$.

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

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

Then $u \mathrel{\mathcal R} v$.

Thus by the definition of transitive closure, there exist $x_0, \dots, x_n \in V$ such that $x_0 \mathrel{A} \dots \mathrel{A} x_n$.

Then $(x_0 \mathrel{A} \dots \mathrel{A} x_n)$ is a directed walk from $u$ to $v$.

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