For two regular languages, why is the set of words from one that don't have a subsequence in the other also regular?
Since this is a homework question, I'll just give a hint.

Hint: Let $L_2$ be the set of supersequences of words in $L_1$, i.e.,

$$L_2 = \{x \in \Sigma^* : \exists y \in L_1 . \text{$y$ is a subsequence of $x$}\}.$$

If $L_1$ is regular, what can you say about $L_2$?  Is it regular?  If you have a NFA for $L_1$, can you construct a NFA that would recognize $L_2$?

If you are not sure, try some examples of regular languages $L_1$, and work out $L_2$ looks like for each example.