Are all $\Sigma^0_1$ sets of infinite sequences infinite?
First observe that any $S \subseteq \{0,1\}^\mathbb{N}$ described by a $\Delta^0_1$-formula is clopen (open and closed subset), where we topologize with the product topology. Indeed, the subbasic sets of the form $B_{i,k} = \{x \in \{0,1\}^\mathbb{N} \mid x(i) = k\}$ are clopen by definition, and then complements (negation), finite unions (disjunction and bounded $\exists$) and intersections (conjunction and bounded $\forall$) of those are still clopen.
A $\Sigma^0_1$ subset $S \subseteq \{0,1\}^\mathbb{N}$ is a countable union of $\Delta^0_1$ sets, therefore it is open. But every open subset of $\{0,1\}^\mathbb{N}$ is either empty, or it contains a basic open set and is therefore uncountable.