Cho $B = \left( {1 - \frac{1}{2}} \right).\left( {1 - \frac{1}{3}} \right).\left( {1 - \frac{1}{4}} \right)....\left( {1 - \frac{1}{{20}}} \right)$.

So sánh B với $\frac{1}{{21}}$.

Ta có

$B = \left( {1 - \frac{1}{2}} \right).\left( {1 - \frac{1}{3}} \right).\left( {1 - \frac{1}{4}} \right)....\left( {1 - \frac{1}{{20}}} \right)$

$B = \frac{1}{2}.\frac{2}{3}.\frac{3}{4}.....\frac{{19}}{{20}}$

$B = \frac{{1.2.3....19}}{{2.3.4....20}} = \frac{1}{{20}}$

Vì 20 < 21 nên $\frac{1}{{20}} > \frac{1}{{21}}$

Vậy $B > \frac{1}{{21}}$.