Do $5^2 > 4.5 $ nên $\frac{1}{5^2}$ < $\frac{1}{4.5}$
Tương tự có $\frac{1}{6^2}$ < $\frac{1}{5.6}$
$\frac{1}{7^2}$ < $\frac{1}{6.7}$
... $\frac{1}{20^2}$ < $\frac{1}{19.20}$
Vậy $ M < $ $\frac{1}{4.5}$ +$\frac{1}{5.6}$ +$\frac{1}{6.7}$+... $\frac{1}{19.20}$
⇒ $M < $ $\frac{1}{4}$ -$\frac{1}{5}$+ $\frac{1}{5}$ -$\frac{1}{6}$ +$\frac{1}{6}$ -$\frac{1}{7}$+...
+ $\frac{1}{19}$-$\frac{1}{20}$
⇒ $M< $ $\frac{1}{4}$ - $\frac{1}{20}$ =$\frac{1}{5}$ (đpcm )