# Pumping lemma for regular languages vs. Pumping lemma for context-free languages

How can I prove the next claim:

If a language $$L$$ meets the pumping lemma for regular languages then $$L$$ meets the pumping lemma for context-free languages?

(Without any pre-condition about the regularity of $$L$$)

The pumping lemma of context-free languages means finding $$x,y,z,v,u$$ such $$w=xvyuz, |vu|>0,|vxu|\le p$$, and for all $$n\in\mathbb N$$, $$xv^nyu^nz$$
The regular pumping lemma means finding only $$v,y,u$$ where $$w=abc,|b|>0,|ab|\le p$$ and $$ab^nc\in L$$. ("a","b","c" here are representing sub-words of w)
So, simply take $$x=a,v=b,y=c,u=\epsilon,z=\epsilon$$