Ta có `: a/( a + b + c ) > a/( a + b + c + d ) , b/( b + c + d ) > b/( b + c + d + a )`
`c/( c + d + a ) > c/( c + d + a + b ) , d/( d + a + b ) > d/( d + a + b + c )`
`⇒ a/( a + b + c ) + b/( b + c + d ) + c/( c + d + a ) + d/( d + a + b ) > a/( a + b + c + d ) + b/( b + c + d + a ) + c/( c + d + a + b ) + d/( d + a + b + c )`
`⇔ a/( a + b + c ) + b/( b + c + d ) + c/( c + d + a ) + d/( d + a + b ) > ( a + b + c + d )/( a + b + c + d ) = 1 ( 1 )`
Lại có `: a/( a + b + c ) < ( a + d )/( a + b + c + d ) , b/( b + c + d ) < ( b + a )/( b + c + d + a )`
`c/( c + d + a ) < ( c + b )/( c + d + a + b ) , d/( d + a + b ) < d/( d + a + b + c )`
`⇒ a/( a + b + c ) + b/( b + c + d ) + c/( c + d + a ) + d/( d + a + b ) < ( a + d )/( a + b + c + d ) + ( b + a )/( b + c + d + a ) + ( c + b )/( c + d + a + b ) + d/( d + a + b + c )`
`⇔ a/( a + b + c ) + b/( b + c + d ) + c/( c + d + a ) + d/( d + a + b ) < ( 2 . ( a + b + c + d ) )/( a + b + c + d ) = 2 ( 2 )`
Từ `( 1 )` và `( 2 )` ta có `1 < a/( a + b + c ) + b/( b + c + d ) + c/( c + d + a ) + d/( d + a + b ) < 2 (` Điều phải chứng minh `)`
