 Every set of reals is continuum-Suslin

Every $A\subseteq {\omega }^{\omega }$ is $2^{{\aleph }_0}$-Suslin. Give each $a\in A$ its own label and use the tree of finite pairs $(a\upharpoonright n,a\upharpoonright n)$, then relabel the first coordinate by an injection into a set of size $2^{{\aleph }_0}$.