Pumping Lemma for Regular Language seems to Fail Let $L = \{ab^ncd \mid n \geq 0\}$.  If we take $p = 5$ and $w = abbcd$ and write 
$w_i = xy^iz$, where $x = abb$, $y=c$, $z=d$, then $w_2 = abbccd$ which is not in&nbsp;$L$.  We conclude that $L$&nbsp;is not regular but it obviously is!

What am I doing wrong?
The pumping lemma says that there exists a decomposition of $w$ to $xyz$ such that $y$ can be pumped.

It doesn't say that for all decompositions of $w$ to $xyz$, $y$ can be pumped.

In your example, a possible decomposition is: 

$x=a$

$y=bb$

$z=cd$