Giải thích các bước giải:
\(
\begin{array}{l}
A = \frac{{\rm{1}}}{{\rm{2}}}.\frac{3}{4}.\frac{5}{6}....\frac{{2011}}{{2012}} \\
B = \frac{2}{3}.\frac{4}{5}.\frac{6}{7}....\frac{{2012}}{{2013}} \\
A.B = \frac{1}{2}.\frac{2}{3}.\frac{3}{4}.\frac{4}{5}....\frac{{2011}}{{2012}}.\frac{{2012}}{{2013}} = \frac{1}{{2013}} \\
Có:\frac{1}{2} < \frac{2}{3} \\
\frac{3}{4} < \frac{4}{5} \\
... \\
\frac{{2011}}{{2012}} < \frac{{2012}}{{2013}} \\
= > A < B \\
= > A.A < A.B \\
= > A^2 < \frac{1}{{2013}} \\
= > A < \frac{1}{{\sqrt {2013} }} \\
\end{array}
\)
