ag.algebraic geometry - Reduced scheme and closed points

Brunoh:

1) If $X$ is a quasi-compact scheme such that $mathscr O_{X,x}$ is reduced for every closed point $x$, then $X$ is reduced. Indeed, let $yin X$. The scheme $overline{{y}}$ is a closed subscheme of $X$, hence is quasi-compact, and non-empty because it contains $y$. It thus has a closed point $x$, which is closed in $X$ as well. Now $mathscr O_{X,y}$ is a localization of $mathscr O_{X,x}$, hence is reduced because so is $mathscr O_{X,x}$ by assumption.

2) Let $k$ be a field and let $v$ be the valuation on $k(X_i)_{iin mathbb Z_{> 0}}$ defined by the composition of the successive discrete valuations provided by the $X_i$'s. Let $X$ be the spectrum of the corresponding valuation ring. Then topologically, $X={x_0,ldots, x_n,ldots}bigcup {x_infty}$ where every $x_i$ specializes to $x_{i+1}$, and where $x_infty$ is the unique closed point (the point $x_0$ is the generic one, and $x_i$ corresponds to the prime ideal generated by $X_i$). Now if you remove $x_infty$ you get an open subscheme $U$ of $X$, without any closed point.
Of course, $U$ is reduced, but $Utimes_k mathrm{Spec}; k[epsilon]$ (with $epsilonneq 0$ and $epsilon^2=0$) is not reduced, and homeomorphic to $U$.