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Given three languages $L_1, L_2, L_3$ with $L_1$ and $L_3$ being semi-decidable and $L_1 \subseteq L_2, L_2 \subseteq L_3$. Can I deduce from these properties, that $L_2$ is also semi-decidable and how would I proof this?

Intuitionally it seems obvious that $L_2$ is also semi-decidable, but I didn't find a way to prove (or falsify) it.

So far I only found information about the closure of semi-decidability under $\cup$ and $\cap$. Can I use these properties in any way?

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No, because every language $L$ satisfies $$ \emptyset \subseteq L \subseteq \Sigma^* $$ and $\emptyset$ and $\Sigma^*$ are both semi-decidable.

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