Inequality in recurrence relation

Question

I'm having a mental block understanding what is probably a simple inequality in a guess and check example for a recurrence relation. Would someone please explain to me how they obtain the inequalities at the very bottom here?

Answer 1

If $n\ge 2^{2/a}$, then

$$\lg n\ge\lg\left(2^{2/a}\right)=\frac2a$$

by definition, so $\lg n\ge\frac2a$. Multiplying both sides by $\frac{a}2n$, we see that $\frac{a}2n\lg n\ge n$. But then

$$\frac{a}2n\lg n+n\le\frac{a}2n\lg n+\frac{a}2n\lg n=an\lg n\;.$$