I am trying to show whether this language is regular or not:

$$L = {0^m1^n mid m neq n}$$

Since I cannot create or think of an automaton that recognizes $L$, I am suspecting that $L$ is not regular. From the book I am using, it seems I can use the pumping lemma, I have done this:

Let $p$ be the pumping lemma constant, and let $w=0^p1^{p+1}$

In this case

$$|w|=2p + 1geq p$$

We can divide $w$ into $xyz$, where $x=0^{p-1}$, $y=0$, $z=^{p+1}$, then $|y|geq 1$

If L was regular, then $forall k geq 0, xy^kz in L$. Let choose $k=2$, then:

$$xyyz=0^{p-1}001^{p+1}=0^{p+1}1^{p+1}$$

Here is where I am stuck, how do I prove that $L$ is not regular?

Another question, pumping lemma applies only to infinite languages?