Question 11
For a function $f : \mathcal{P}(\mathbb{N}) \to \mathbb{N}$ and a subset $A \subseteq \mathbb{N}$, we say $f$ is *A-predictor* if the set $\{x \in \mathbb{N} \mid x \notin A, f(A \cup \{x\}) \neq x\}$ is finite. Prove that there exists a function that for every subset $A$ of natural numbers is *A-predictor*.