 Uniform holomorphic convergence without boundary derivative convergence

The entire functions $g_n(z)=z^n/n$ converge uniformly to zero on the closed unit disc, because their supremum norm is $1/n$. Their derivatives $g_n^{\prime }(z)=z^{n-1}$ instead have supremum norm one there, and equal one at $z=1$. On each smaller disc of radius $0<r<1$, the derivative supremum is $r^{n-1}\to 0$. Thus the Weierstrass convergence theorem controls derivatives on compact subsets of an open domain, not automatically on the boundary of a closed set on which function convergence is known.