Problem Prove that the language $L = \{a^nb^kc^{n + k} : n \geq 0, k \geq 0\}$ is not regular by Pummping Lemma.
Pumping Lemma Let $L$ be an infinite regular language. Then there exists some positive integer $m$ such that any $w \in L$ with $|w| \geq m$ can be decomposed as: $w = xyz$, with $|xy| \leq m$, and $|y| \geq 1$ such that $w_i = xy^iz,$ is also in $L$ for all $i = 0,1,2 \ldots$
Note:$|w|$ means the length of string $w$.
My attempt to this problem is as follows,
Let $m$ be any positive integer, we definie $w$, $w = a^mb^mc^{m+ m}$. Since $m > 0$ which implies $n \geq 0$ and $k \geq 0$ are true. Hence, $w \in L$.
Now consider $|xy|$, by Pumping Lemma we know that $|xy| \leq m$ which implies that $xy$ must consists of entirely a's. Furthermore, $|y| \geq 1$, hence $y = a^k$ for some positive integer $k$.
Assuming $k = 1$ to consider $w_0 = xy^0z = a^{m-1}b^{m}c^{2m}$, this string is obviously not in $L$ because $m - 1 + m \neq 2m$. Similarly for $k > 1$, we can always use $y^0$ to extract some a's from $a^m$ which contradicts the constraints $n + k$. Therefore $L$ is not regular $\square$.
Any suggestion or comment would be greatly appreciated. Thank you.