PomeLemma 8.11.5.9416072D448F Let $f_ j$ be uniformly bounded smooth local $m$-subharmonic functions with $f_ j\downarrow f$, and fix a smaller concentric ball $V\Subset U$. There are uniformly bounded smooth local functions $v_ j\downarrow v$ which equal $f_ j$ on a neighborhood of $\overline V$ and equal one fixed strictly plurisubharmonic $\psi $ on one fixed boundary collar. Moreover $v=f$ near $\overline V$ and $v=\psi $ on that collar.
Lemma 8.11.59416072D448F
Proof. Write $U=\{ |z|<R\} $ and $|f_ j|\leq M$. Take $\psi =A(|z|^2-R^2)+b$, with $b>M+2$ and then $A$ so large that $\psi <-M-2$ on a neighborhood of $\overline V$. It exceeds $M+2$ on a fixed sufficiently thin boundary collar.
Use the regularized maximum $M_\epsilon $ constructed in 8.2.9. Choose $0<\epsilon _ j\downarrow 0$ with $\epsilon _ j<1$, and put $v_ j=M_{\epsilon _ j}(f_ j,\psi )$. The lemma preserves the local unshifted cone condition. Its two monotonicities give $v_ j\downarrow \max (f,\psi )=v$, and the fixed gaps give the two asserted regions of exact equality. The bounds
\[ \max (f_ j,\psi )\leq v_ j\leq \max (f_ j,\psi )+\epsilon _ j/2 \]
give a common bound. In particular all differences $v_ k-v_ j$ have support in one fixed compact subset of $U$. $\square$
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