Exercise 3, § 1

In the product ring $A = \mathbf{R}^{\mathbf{N}}$, let (foreach $n \in \mathbf{N}$) $m_n$ be the maximal ideal consisting of the $f : \mathbf{N} \to \mathbf{R}$ such that $f(n) = 0$. Show that $a = \mathbf{R}^{(\mathbf{N})}$ is an ideal of A contained in the union of the $m_n$ but in none of these maximal ideals.