Discrete Harmonic functions converge to a harmonic function

Question

I am having a problem about Jost's PDE book (3rd edition).

Let $\Omega \subset \mathbb{R}^d$ be a domain (i.e., open and bounded). We consider the orthogonal grid of mesh size $h\mathbb{Z}$ ($h>0$), and let $\overline{\Omega}_h = \Omega \cap h\mathbb{Z}$. Denote $\Omega_h$ to be the set of interior vertices of $\overline{\Omega}_h$ and $\Gamma_h$ be the set of boundary vertices. Here, an interior vertex is an element in $\overline{\Omega}_h$ such that its nearest vertices are also in $\overline{\Omega}_h$; otherwise it is a boundary vertex.

In the book, two lemmas are established:

1. Discrete maximum principal: If $\Delta_h u^h \ge 0$ in $\Omega_h$ where $\Omega_h$ is discretely conncected, then the maximum of $u^h$ must occur on $\Gamma_h$. Here, $\Delta_h$ is the discrete Laplacian via the stencil $$\frac{1}{h^2} [ 1, -2, 1]$$ in each variable.
2. Suppose in $\Omega_h$, $$\Delta_h u^h = f^h$$ Let $x_0 \in \Omega_h$ and suppose $x_0$ and all its neighbor have distance greater or equal to $R$ from the boundary $\Gamma_h$. Then $$|u^h_\iota(x_0)| \le \frac{d}{R}\max_{\Omega_h}|u^h| + \frac{R}{2}\max_{\Omega_h}|f^h|$$ where the subscript $\iota$ denotes the symmetric difference quotient.

Theorem 4.1.2 states that

If all solution $u_h$ of $$\Delta_hu^h = 0 \text{ in } \Omega_h$$ are bounded independently of $h$ (i.e., $\max_{\Gamma_h}|u^h|\le \mu$), then in any subdomain $\Omega' \subset \subset \Omega$, some subsequence of $u^h$ converges to a harmonic function as $h\to 0$. Convergence means $$\lim_{n \to 0} \max_{x\in \Omega_n} |u_n(x) - u(x)| = 0$$

The book does not talk very much about this theorem. In fact, after proving the second lemmas and mentioning that we can bounded the difference quotient of any order, the book then says that we obtain Theorem 4.1.2.

First, I am wondering how to find this harmonic $u$. Because the discrete harmonic function $u^h$ is defined only on the grid $h\mathbb{Z}$ which depends on $h$, in general, it does not make sense to say $u^{h_1}(x) - u^{h_2}(x)$. The usual technique to find the limiting function does not work. Besides, I cannot see the relation between Theorem 4.1.2 and the above two lemmas. I suppose that the uniform bound of the solutions $u^h$ should be used to find a convergent subsequence, but I struggled with giving a proof.