Construction of Projective Space (Part 2)

The Proj construction allows us to define the projective space $\mathbb{P}^n(k)$ in a new way, namely as $\text{Proj}k[x_0, \ldots, x_n]$. One advantage of this approach is that it emphasizes the symmetry of $\mathbb{P}^n(k)$: we glue all basic opens $D_+(f)$ instead of just $D_+(x_i)$, as in the previous approach. In particular, in this new construction, it is easy to see that $D_+(f)$ are affines, whereas in the previous construction, the corresponding sets are not clearly affine (except for $D(x_i)$.

We now show that these two constructions agree.