Using Pumping Lemma prove that the language $L = \{a^ib^j \mid i,j \in N \}$ is Not Regular.

Using Pumping Lemma prove that the language $L = \{a^ib^j \mid i,j \in N \}$ is Not Regular.

Proof:

Assume that $L$ is Regular.Pumping Length = $P$. We choose $w = a^{P-2}b^{P+2} \in L$

We divide $w$ in $xyz$.$x= a, y = a^{P-3}bb, z = b^P$

Now we Pump with $i = 2$. $xy^iz = xy^2z$ = $aa^{P-3}bba^{P-3}bbb^P$ = $a^{P-2}bba^{P-3}b^{P+2} \notin L$. QED

Did I do it right?