2.5: Let u be harmonic and nonnegative, show that u is constant. (Hint use the previous exercise).
The previous exercise was posted in another question, stated the following.
2.4: Let $u:B(0,R)\subset \mathbb{R^d}\rightarrow\mathbb{R}$ be harmonic and nonnegative. Prove the following version of the Harnack inequality: $$\dfrac{R^{d-2}(R-|x|)}{(R+|x|)^{d-1}}u(0)\leq u(x)\leq \dfrac{R^{d-2}(R+|x|)}{(R-|x|)^{d-1}}u(0)$$
I used the poisson's integral for the ball that states the following.
$$u(x)=\dfrac{R^2-|x|}{n\alpha(n)R}\int_{\partial B(0,R)}\dfrac{g(y)}{|x-y|^n}dS(y)$$ and the fact that for $y\in \partial B(0,R)$, $|y|=R$ and $|x|-|y|\leq |x-y|\leq |x|+|y|$ that is $|x|-R\leq |x-y|\leq |x|+R$.
Any hints on how to apply 2.4 to 2.5? thanks in advance.
Just let $R\to\infty$. Both sides of the inequality converge to $u(0)$.