https://www.math.ucdavis.edu/~emsilvia/math127/chapter8.pdf
In the last part of the proof of Theorem 8.2.3, why does the inequality $$|f_{M^\ast+1}(z)-f(z)|<\epsilon/2$$ hold uniformly for all $z$? A few lines above it was noted that this inequality holds for a fixed $z$, but this line is supposed to prove uniform convergence.
The author notices that it follows from the fact that $n, z$ were arbitrary, but I don't see how this helps. Yes, the above inequality holds for arbitrary $z$, but $M^\ast$ depends on $z$.
$M^*$ is allowed to depend on $z$, but note that the final result doesn’t depend on $M^*$ (perhaps better notation would be $M^*(z)$), so we can let $M^*$ vary with $z$.