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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 $L$)

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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 $x=a,v=b,y=c,u=\epsilon,z=\epsilon$

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