I want to prove (or disprove) the following conjecture:
Let $\Sigma$ be a finite alphabet, and let $L \subseteq \Sigma^*$ satisfy that there are no $v, w \in L$ such that $v$ is a proper substring of $w$. Then $L$ is finite.
I have the intuition that this is true, haven't been able to find a counter example, but am unable to prove it. If this is true, how can we prove it? If this is false, how can we construct such infinite $L$?