Application of fusion technique to the solution of Harrington problem and its generalizations to Baire functions, part I

In this paper we provide solutions of the Harrington problem (along with a few generalizations) proposed in a book Analytic Sets. The original problem asks if for arbitrary sequence of continuous functions from ${\mathbb{R}}^{\omega }$ to a fixed compact interval we can find a subsequence point-wise convergent on some product of perfect subsets of $\mathbb{R}$. We reduce aforementioned problem to functions from $C^{\omega }$ to $C$, where $C$ is a standard Cantor set as well as also provide solution to the problem with Baire functions in place of continuous ones. Our main focus is on showing applications of the fusion lemma - a result about perfect trees used among others to prove minimality of Sack’s forcing - to the problem at hand.