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 \(B > \frac{1}{{21}}\).