nth Veronese subring embedding induces isomorphism of schemes (Vakil FOAG 6.4.D)

Question

Exact exercise is: show the embedding map of graded rings $S_{n\bullet} \to S_\bullet$ for $S_\bullet$ finitely generated graded ring induces an isomorphism $\operatorname{Proj} S_\bullet \to \operatorname{Proj}S_{n\bullet}$. The hint is to show that we can set up isomorphisms of $D(f) \to D(f)$ for $f \in S_+$ homogeneous degree $n$, which is sufficiently clear by considering zero components of localized rings and by considering that $D(f)$ cover $\operatorname{Proj}S_\bullet$ (which is simply by considering that the proj construction starts by taking sets of primes and so for $f_i$ generators of $S_\bullet$ we can see $D(f_i^n) = D(f_i)$). So far, have successfully done everything up to showing that we get a sheaf isomorphism (meaning considering the schemes to be sheaves on the same space via the inverse map on open sets gives an isomorphism of sheaves) since the isomorphisms of the distinguished opens give isomorphisms of stalks. However, it still remains to show that the induced map is a homeomorphism (i.e. that it is a set isomorphism with a continuous inverse). I get stuck here, since I think the isomorphism induces on distinguished opens does not necessarily mean that the map as a whole is a set isomorphism with a continuous inverse. I'd love any hints on how to wrap this up.