Let $L$ be the set of countable limit ordinals. For each $\alpha \in L$, let $\langle \alpha_n : n < \omega \rangle$ be a strictly increasing cofinal sequence in $\alpha$. Define a linear order on $L$ by $\alpha \prec \beta$ iff the least $n < \omega$ for which $\alpha_n \neq \beta_n$ satisfies $\alpha_n < \beta_n$ (So $\prec$ is just the lexic order). I am trying to show the following:
$L$ is not a countable union of well ordered subsets under $\prec$.
Every uncountable suborder of $L$ contains a copy of $\omega_1$.
I don't see how to show any of these. Any help/hints would be appreciated. I should add that this is not a homework question - A mean friend gave this problem to me.