n-dim Projective Space is NOT affine if n > 0

In the previous post, we glue together affine spaces $X_i = \text{Spec} A_i = \text{Spec} k\left[x_0, \ldots, x_n, \frac{1}{x_i} \right]$ to get $\mathbb{P}^n(k)$.

To show that $\mathbb{P}^n(k)$ is not affine, we first compute its global sections.

The global sections of $\mathbb{P}^n$ are in bijective correspondence with tuples $(s_0, \ldots, s_n) \in A_0 \times \cdots \times A_n$ that are compatible with respect to restrictions. However as restrictions $A_i \to A_{ij}$ are simply inclusion maps, we have $(s_i)_{X_{ij}} = (s_j)_{X_{ij}} $ iff $s_i = s_j$ as elements of $A_{ij}$. So we have
$$s_0 = s_1 = \ldots = s_n \text{ as elements of } k(x_0, \ldots, x_n).$$
Thus the global sections $\mathbb{P}^n$ is $\bigcap_i A_i = k$. If $\mathbb{P}^n(k) $ is affine, then it must then be $\text{Spec} k$. But if $n >0$, then $\mathbb{P}^n(k)$ consists of more than one point. In fact, its closed points are in bijective correspondence with points of the classical projective  Space.