A short proof of the Segal conjecture for $C_p$

We give a new proof of the generalized Segal conjecture for the cyclic group of prime order p, i.e. the fact that the Tate diagonal of any bounded below spectrum is a p-completion. The idea, inspired by Hahn and Wilson's argument, is to use descent along the equivariant map from the p-fold tensor power of the field with p elements to its topological Hochschild homology. The only computational input that we use is the calculation of the homotopy rings of this topological Hochschild homology and its Tate construction, together with the map induced by the cyclotomic Frobenius.