Let L$1=\{0$n$1$m $|$ n $6=$ m$\},$ and consider the language L$$

$1.$

$($a$)$ Here is a wrong $$proof$$ that L$$

$1$ is not regular:

First, we proved in lecture that the class of regular languages is closed un$-$

der the star operation. Second, we proved in Handout $4$ that the language

L$1$ is not regular. Therefore, by closure properties, we conclude that L$$

$1$

is not regular.

Which of the step$($s$)$ above are wrong? $($first$,$ second, and$/$or conclusion?$)$ Provide

a short justification$/$explanation$.$

$($b$)$ Prove that L$$ is regular.