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

Question

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?

Answer 1

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