Prove that every substring closed language L ⊆ {0, 1}* is regular

For $x,y \in \{0,1\}^$ a language $L ⊆ \{0, 1\}^$ is called substring closed, if $y \in L$ and $x \preceq y$ ($x$ substring of $y$) implies $x \in L$.

I want to prove that every substring closed language $L ⊆ \{0, 1\}^*$ is regular.

Is it enough to have a $L$ such that just $x$ in it, then prove that $L$ is regular?

There are some substring-closed languages that are not regular.

Here is an example. Let $C=\{01^n0^n1\mid n\ge1\}$ and $F=\{f\mid \exists c\in C, f\preceq c\}$, i.e., $F$ is the language of all substrings of strings in $C$.

• $F$ is substring-closed, since a substring of a substring of string $c$ is also a substring of $c$.

• The intersection of $F$ and the regular language $\{01w01\mid w\in \{0,1\}^*\}$ is $C$, a non-regular language. So $F$ is not regular.

StackExchange Computer Science Q#149336, answer score: 16