I have the following problem:
Let X be a Noetherian Scheme and suppose that $X_{red}$ is affine. Show that this implies that X is affine.
OK, so I know the "classical" proof of this using Serre's criterion for affineness and with cohomology. However, I encountered this in an early chapter of Görtz-Wedhorn's book where none of these concepts have thus far been defined. I have been trying to come up with an elementary proof but without much success. Clearly, we can assume that the ideal sheaf $\mathcal{N}$ satisfies $\mathcal{N}^2 = 0$.
I would be very grateful for help with this problem of any sort, ranging from hints to solutions.