Is there a linear order with this property

Question

I was trying and failing to construct a linear order L each of whose uncountable subsets contains an uncountable well ordered subset but L is not a countable union of well ordered subsets. Is this possible?

Thanks for any ideas!

Answer 1

For each countable limit ordinal $\alpha$ choose some $\langle \alpha_n : n < \omega \rangle$ increasing and cofinal in $\alpha$. Then it can be checked that the set $L$ of these sequences ordered lexicographically satisfies your requirements.