Let $f_k$ be a function from the finite set $S_k$ to the real interval [0,1], with $S_k\subseteq S_{k+1}$. Let also $S=\bigcup_{k\ge 1} S_k$ and assume that $S$ is the set of rational numbers in [0,1].
If $f_k(x)$ converges pointwisely to the zero function $f(x)=0$, with $f:S\to[0,1]$, and $0\le f_{k+1}(x)\le f_{k}(x)$, prove or disprove that the convergence is actually uniform over $S$.
Note. There might be the answer in the theorem reported here. Unfortunately, I don't have access to Spivac's book and it's not clear to me whether I can apply it (for instance, $S$ is not compact).
Consider $$ f_k(x)=\left\{\begin{array}{} x^k&\text{if }x\ne1\\ 0&\text{if }x=1 \end{array}\right. $$ $f_k$ converge pointwise to $0$ and $0\le f_{k+1}(x)\le f_k(x)$, yet the convergence is not uniform.