Ta có:

1√2005+√2004=√2005–√2004(√2005+√2004)(√2005–√2004)12005+2004=2005–2004(2005+2004)(2005–2004)

=√2005–√20042005–2004=√2005–√2004(1)$=2005–20042005–2004=2005–2004(1)$

Ta có:

1√2004+√2003=√2004–√2003(√2004+√2003)(√2004–√2003)12004+2003=2004–2003(2004+2003)(2004–2003)

=√2004–√20032004–2003=√2004–√2003(2)=2004–20032004–2003=2004–2003(2)

Vì √2005+√20042005+2004 > √2004+√20032004+2003 nên:

1√2005+√2004≤1√2004+√200312005+2004≤12004+2003 (3)

Từ (1),(2),(3) suy ra: 

√2005–√20042005–2004 < √2004–√2003