# Decidability of $\{⟨G⟩ \mid \text{$G$is CFG and$L(G) ⊈ \Sigma^+$}\}$

I want to prove that the following language is decidable:

$$\mathit{SEQ}_{\mathit{CFG}} = \{⟨G⟩ \mid \text{G is CFG and L(G) ⊈ L}\}, \text{ where } L = \Sigma^* - \{\epsilon\}$$

So, I think about the relationship between equality and substrings.

for example, I saw in different questions that when we have $$L(G) = L(H)$$ we can perform that proof on the $$L(G) ⊆ L(H)$$.

So can I do that with $$L(G) ⊈ L$$?

So what is the benefit of saying $$L = \Sigma^* - {\epsilon}$$ ? I'm a bit confused. Can anyone help me with this question?

The idea is that $$L(G) \not\subseteq \Sigma^* \setminus \{\epsilon\}$$ iff $$\epsilon \in L(G)$$.