Chains of Well-ordered sets

Question

Let $\{X_i\mid i\in I\}$ be a chain of sets and $\{R_i\mid i\in I\}$ be the corresponding chain of relations such, $R_i$ well orders $X_i$. Show that $\bigcup_{i\in I}R_i$ well orders $\bigcup_{i\in I}X_i$, that is for any $Y\subset\bigcup_{i\in I}X_i$, there is a $y'\in Y$ such that $(y',y)\bigcup_{i\in I}R_i$, for all $y\in Y$. Any help greatly appreciated. Thanks

Answer 1

Let $I=\mathbb Z$, $X_i=\{\,k\in\mathbb Z\mid k+i>0\,\}$, $R_i$ the standard order on $X_i$ (which is a well-order as $X_i$ is just a translated $\mathbb N$). Then $\bigcup R_i$ is just the standard order on $\bigcup X_i=\mathbb Z$ and is not a well-order.