  We prove the undecidability of MSO on $\omega$-words extended with the
second-order predicate $U_1(X)$ which says that the distance between
consecutive positions in a set $X \subseteq \mathbb{N}$ is unbounded. This is
achieved by showing that adding $U_1$ to MSO gives a logic with the same
expressive power as $MSO+U$, a logic on $\omega$-words with undecidable
satisfiability. As a corollary, we prove that MSO on $\omega$-words becomes
undecidable if allowing to quantify over sets of positions that are ultimately
periodic, i.e., sets $X$ such that for some positive integer $p$, ultimately
either both or none of positions $x$ and $x+p$ belong to $X$.
