Proof. By definition, the ordering is reflexive, antisymmetric, translation-invariant, and invariant under scaling by positive constants. It remains to show that $\le$ is transitive, or equivalently, the sum of two positive elements is positive.

Let $x, y \in A$ be positive, then $x + y$ is self-adjoint. Thus there exists $\lambda \ge \norm{x}_{A}$ and $\mu \ge \norm{y}_{A}$ such that $\norm{\lambda - x}_{A} \le \lambda$ and $\norm{\mu - y}_{A} \le \mu$, so $\norm{(\lambda + \mu) - (x + y)}_{A} \le \lambda + \mu$, and $x + y$ is positive by Proposition 34.8.3.$\square$