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What is the result of $1\div 1\frac{2}{2001}\div 1\frac{2}{2003}\cdots \div 1\frac{2}{2023}$? 
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$1\div 1\frac{2}{2001}\div 1\frac{2}{2003}\cdots \div 1\frac{2}{2023}$  $=1\div \frac{2003}{2001}\div \frac{2005}{2003}\cdots \div \frac{2025}{2023}$  $=1\times \frac{2001}{2003}\times \frac{2003}{2005}\cdots \times \frac{2023}{2025}$  $=\frac{2001}{2025}$ 