In the book "A Course in $p$ Adic Analysis" by Alain M. Robert, he presents the following theorem.
"Equipped with the $p$ adic metric, $\mathbb{Z}_p$ forms a topological ring that is complete, compact, and metrizable."
The proof of this fact is the following: Since $\mathbb{Z}_p$ is a topological group under addition (proven before in text), it suffices to show that multiplication is continuous. Let $x=a+h$ and $y=b+k$ be elements of $\mathbb{Z}_p$ with fixed $a,b\in\mathbb{Z}_p$. Thus, $$|xy-ab|=|(a+h)(b+k)-ab|=|ak+bh-hk|\le\max(|a|,|b|)(|h|+|k|)+|k||h|$$ which tends to $0$ as $|h|,|k|\to 0$. This proves continuity in $\mathbb{Z}_p\times \mathbb{Z}_p$.
However, I am confused on the final inequality. Specifically, how did he get from $$|xy-ab|=|(a+h)(b+k)-ab|=|ak+bh-hk|?$$ Using basic arithmetic I thought it should be written as $$|xy-ab|=|(a+h)(b+k)-ab|=|ak+bh+hk|=|(ak+hk)+bh|=|kx+bh|\le\max(|kx|,|bh|)?$$ Doesn't this provide the same conclusion since this tends to $0$ as $|h|,|k|\to 0$? Am I making a mistake somewhere?