# Is the set of languages satisfying the pumping lemma closed under concatenation?

Let $$L$$ be the set of all languages that satisfy the pumping lemma, including non-regular languages that satisfy it. Is the set $$L$$ closed under concatenation?

I couldn’t prove it or find a counterexample.

Suppose that $$L_1$$ satisfies the pumping lemma: there exists $$p_1$$ such that every word $$w \in L_1$$ of length at least $$p$$ can be decomposed as $$w = xyz$$, where $$|xy| \leq p_1$$, $$y \neq \epsilon$$, and $$xy^iz \in L_1$$ for all $$i \geq 0$$. Similarly, suppose that $$L_2$$ satisfies the pumping lemma, with constant $$p_2$$.
I claim that $$L = L_1L_2$$ satisfies the pumping lemma with constant $$p = p_1 + p_2$$. Suppose we are given a word $$w = w_1w_2 \in L$$ of length at least $$p$$, where $$w_1 \in L_1$$ and $$w_2 \in L_2$$. Then either $$|w_1| \geq p_1$$ or $$|w_2| \geq p_2$$. You take it from here.