I have a problem with the proof of the Proposition 5 in page 6 of the article https://arxiv.org/pdf/1308.5918.pdf. Until equation $(3.5)$ all is clear, but when the author says $"$By (3.1), strictness of the maximum of $u-\psi$ implies that there is a $k > 0$ small such that by sliding $\psi$ downwards to some $\psi-k$, we have $$ \Omega^+:=\{u-\psi+k>0\}\subseteq\mathbb{B}_r(x) $$ and also $$ u = \psi-k\ \ \text{on}\ \partial\Omega^+." $$ I don't understand why $\Omega^+$ is contained in $\mathbb{B}_r(x)$. Some suggestions or hints to this aim?
Thank You