Using basic definition, we show that $n^2 - 10n = \Omega(n^2)$.
For, $n \geq \frac{n}{2}$ for $n \geq 0$
$n – 10 \geq \frac{n}{2 \cdot 10}$ for $n \geq 10$
$n^2 - 10n \geq \frac{n^2 }{ 20}$ for $n \geq 10$
$ n^2 - 10n \geq c \cdot n^2$ for $n \geq n_0$ where $c= \frac{1}{20}$ and $n_0 = 10$.
Therefore, by basic definition, $n^2 - 10n = \Omega(n^2)$.
I don't understand how this inequality was derived: $n – 10 \geq \frac{n }{2 \cdot 10}$.