I am studying Theorem 2.9 under the section of Convergence Theorems from Gilbarg and Trudinger's book Elliptic Partial Differential Equations of Second Order. Here I have found some difficulty. I have understood everything before the inequality
$$\underset {x \in \Omega'} {\mathrm {sup}}\ |u_m (x) - u_n (x)| < C\epsilon.$$ But why will this hold? I know Harnack's inequality. If we apply this inequality here we obtain
$$\underset {x \in \Omega'} {\mathrm {sup}}\ |u_m (x) - u_n (x)| < C \underset {x \in \Omega'} {\mathrm {inf}}\ |u_m (x) - u_n (x)|.$$ But how does this imply the above inequality? This can only happen if $$\underset {x \in \Omega'}{\mathrm {inf}}\ |u_m (x) - u_n (x)| \le \epsilon.$$ If $y \in \Omega'$ then we are done. But if it is not how do we proceed? Please help me in this regard.
Thank you in advance.