Open Subset of Affine NOT Quasi-compact

Let $X = \text{Spec} k[x_1, x_2, \ldots].$ Let $\mathfrak{m}$ be the maximal ideal $(x_1, x_2, \ldots).$. Let $U = X - [\mathfrak{m}].$ Then $U = \bigcup_{i} D(x_i)$.

Claim: $U$ is not quasi-compact

Proof.
 Suppose $U$ can be covered by finitely many of the $D(x_i)$'s say $U= \bigcup_{i=1}^n D(x_i)$. Then $(x_1, \ldots, x_n) \in U$ but $(x_1, \ldots, x_n) \not \in D(x_i)$, a contradiction.