Prove language is regular let's have these two languages in the alphabet $\{a,b,c\}$:

$L_1 = \{ w \mid w \text{ is a palindrome and $|w| &lt; 200$}\}$

$L_2 = \{ w \mid w \text{ is a suffix of $u$ and $|u|$ is a prime number and $u$ doesn't contain a $c$}\} $

and I have to prove that $L_1 ∪ L_2$ is regular.

So my idea is I have to prove both languages are regular and therefore their union would be regular. $L_1$ is obviously regular since it's finite. I also know that the language consisting of all suffixes of words from a regular language is also regular, so I narrowed it down to proving that 

$L = \{w \mid |w| \text{ is a prime number and $w$ doesn't contain a $c$}\}$

any ideas on that one? Thanks in advance.
The language $L$ is not regular, but $L_2$ is. In fact $L_2 = \{a,b\}^*$.
Indeed if $w \in L_2$, then by definition, $w$ is suffix of a word of $\{a,b\}^*$ and thus $w \in \{a,b\}^*$. Thus $L_2 \subseteq \{a,b\}^*$. To prove the opposite inclusion, consider a word $w \in \{a,b\}^*$. Let $p$ be a prime such that $p \geqslant |w|$ and let $u = a^{p-|w|}w$. By construction, $|u| = p$,
$u \in \{a,b\}^*$ and $w$ is a suffix of $u$, and thus $w \in L_2$.