Cho a, b, n thuộc ℕ. Hãy so sánh  $\frac{a+n}{b+n}$ và  $\frac{a}{b}$.

Xét  $\frac{a+n}{b+n}-\frac{a}{b}=\frac{b\left(a+n\right)}{b\left(b+n\right)}-\frac{a\left(b+n\right)}{b\left(b+n\right)}=\frac{ba+bn-ab-an}{b\left(b+n\right)}=\frac{bn-an}{b\left(b+n\right)}$

Nếu a ≤ b thì b – a ≥ 0

Suy ra  $\frac{a+n}{b+n}-\frac{a}{b}\ge 0$

Do đó  $\frac{a+n}{b+n}\ge \frac{a}{b}$

Nếu a ≥ b thì b – a ≤ 0

Suy ra  $\frac{a+n}{b+n}-\frac{a}{b}\le 0$

Do đó  $\frac{a+n}{b+n}\le \frac{a}{b}$.