How to prove that $\{0^n 1^{5n} \mid n \ge 10000 \}$ is not a regular language?
The condition $n&gt;10\,000$ makes no real difference. If the pumping length is greater than $10\,000$, your existing proof works. If the pumping length is $10\,000$ or less, then pumping the string $0^{10\,000}1^{50\,000}$ gives you a string that isn't in the language because it has too many zeroes.