Confused about pumping lemma, What i'm missing?
you can't claim to know how $w$ distributes.

you know that some $w \geq n_0$ exists, in such a way it can be denoted as 

$w=xyz$

$y \neq \epsilon $

$ |xy|\leq n_0 $

and for every $i&gt;0: xy^iz \in L$

your error is that you claimed you know the distribution of $w$ to $x,y,z$. Namely, you fixed $y$ to be $10$

The language of course can be pumped, since it's regular. But for any $w = xyz, y$ must be a word of only $0$, since $1's$ cannot be pumped.