Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = - \Delta$, ie $u$ harmonic.

Then we can prove the comparison principle stated in STEP 1 below (Caffarelli, 1998 p. 40):

$$u\big|_{B^+_{1/2}} \le M.$$

I've some questions on the proof below (PROOF OF STEP 1).

In particular, what does the

**(c)**observation mean? What does the author mean by $M>M_0$ large enough? What is the "absurde assumption" that will lead to a contradiction?Why after proving that

"if $M \ge M_0$ large, we can construct a sequence of points, $X_k$ all contained in $B^+_{3/4}$, $X_k → \{x_n = 0\}$, and such that $u(X_k)$ goes to $+\infty$"

do we get that the proof of STEP 1 is complete?