Let $A$ be DFA and $B$ be DFA. Let $K$ be an alphabet and $L(A) \subset K^*$ and $L(B) \subset K^*$ Prove that there exists such $p \in \mathbb{N} $ that, if for every $w \in K^*$ where $|w| \le p $ and $$ w \in L(A) \Rightarrow w \in L(B) $$ is true then $$L(A) \subset L(B)$$
My intution is using a pump lemma and I suppose that $p$ should be length of pump for $L(B) $ . Please give me help hand.
Hint. Let $L = L(A) - L(B)$. Then $L$ is a regular language and $L(A) \subseteq L(B)$ if and only if $L = \emptyset$. Furthermore, saying that for all $w \in K^*$ such that $|w| \leqslant p$, $$w \in L(A) \implies w \in L(B)$$ is equivalent to saying that all the words of $L$ have length $> p$. Thus it remains to prove that if $L$ is a regular language, there exists $p \in \mathbb{N}$ such that, if $L$ contains no word of length less than $p$, then $L$ is empty.