Prove that $L=\{a^n b^l : n \leq l\}$ is not regular by pumping lemma

Question

I'm currently trying to prove that $L=\{a^n b^l : n \leq l\}$ is not regular by pumping lemma

My proof:

If we choose $w$ such that $w=a^P b^P$, then since $|xy| \leq p$, $y$ must be $a^P$, meaning it can be pumped any number of times $i$, such that $a^{iP} b^P \in L$. Since $iP \leq P$ is not true for any $i > 1$, it cannot exist in $L$, therefore $L$ cannot be regular.

Is this a valid proof? I don't fully comprehend the pumping lemma, so please let me know if/where I may be going wrong.

Thank you

Answer 1

There is a bit of a confusion on the notations:

• what is $P$? If it is the pumping length, then say it so;
• what is $p$? Is it a typo and should be a $P$?
• what are $x$ and $y$? Don't hesitate to remind the hypotheses of the pumping lemma;
• why would $y$ be $a^P$? The only conclusion you can make is that $y = a^k$ for $1\leqslant k \leqslant P$.

The ideas of the proof are good but you should improve the details.