Ta có:
$(\sqrt[]{2017}+\sqrt[]{2020})^2$
$=2017+2020+2\sqrt[]{2017.2020}$
$=4037+2\sqrt[]{2017.(2018+2)}$
$=4037+2\sqrt[]{2017.2018+2017.2}$
$(\sqrt[]{2018}+\sqrt[]{2019})^2$
$=2018+2019+2\sqrt[]{2018.2019}$
$=4037+2\sqrt[]{2018.(2017+2)}$
$=4037+2\sqrt[]{2018.2017+2018.2}$
Vì $(\sqrt[]{2017}+\sqrt[]{2020})>0$, $(\sqrt[]{2018}+\sqrt[]{2019})>0$ và $(2017.2018+2017.2)<(2018.2017+2018.2)$ nên $\sqrt[]{2017}+\sqrt[]{2020}<\sqrt[]{2018}+\sqrt[]{2019}$.
