SCHWEITZER 2020 Problem 1

We say that two sequences $x,y \colon \mathbb{N} \to \mathbb{N}$ are completely different if $x_n \neq y_n$ holds for all $n\in \mathbb{N}$ . Let $F$ be a function assigning a natural number to every sequence of natural numbers such that $F(x)\neq F(y)$ for any pair of completely different sequences $x$ , $y$ , and for constant sequences we have $F \left((k,k,\dots)\right)=k$ . Prove that there exists $n\in \mathbb{N}$ such that $F(x)=x_{n}$ for all sequences $x$ .