\(\frac{2001}{2000}-\frac{2002}{2001}=\frac{2001.2001}{2000.2001}-\frac{2002.2000}{2000.2001}\)
\(=\frac{\left(2002-1\right).\left(2000+1\right)-2002.2000}{2000.2001}\)
\(=\frac{2002.\left(2000+1\right)-\left(2000+1\right)-2002.2000}{2000.2001}\)
\(=\frac{2002.2000+2002-2000-1-2002.2000}{2000.2001}\)
\(=\frac{2002.2000+1-2002.2000}{2000.2001}\)
\(=\frac{1}{2000.2001}\)