Here is the proof :
d. Liouville's Theorem. We assert now that there are no nontrivial bounded harmonic functions on all of $\Bbb R^n$.
THEOREM 8 (Liouville's Theorem). Suppose $u:\Bbb R^n\to\Bbb R$ is harmonic and bounded. Then $u$ is constant.
Proof. Fix $x_0\in\Bbb R^n$, $r\gt0$, and apply Theorem 7 on $B(x_0,r)$: $$\eqalign{ |Du(x_0)|&\leq\dfrac{\sqrt nC_1}{r^{n+1}}\lVert u\rVert_{L^1(B(x_0,r))}\\&\leq\dfrac{\sqrt nC_1\alpha(n)}{r}\lVert u\rVert_{L^\infty(\Bbb R^n)}\to0,}$$ as $r\to\infty$. Thus $Du\equiv0$, and so $u$ is constant.
I don't know why he can change L1 with the ball $B(x_0,r)$ to L infinity with Rn