由算术-调和平均不等式。
5/(x+y+1) = 5[1/(x+y+(1/3)+(1/3)+(1/3)] <= (5/25)[1/x + 1/y + 3 + 3 + 3]
=(1/5)(1/x + 1/y) + 9/5
代入右边，移项化简。不等式的证明归结于证明
1/x + 1/y + 1/z >= 9.
LHS >= 3/cuberoot(xyz) >= 3/((x+y+z)/3) = 9.
5/(x+y+1) = 5[1/(x+y+(1/3)+(1/3)+(1/3)] <= (5/25)[1/x + 1/y + 3 + 3 + 3]
=(1/5)(1/x + 1/y) + 9/5
代入右边，移项化简。不等式的证明归结于证明
1/x + 1/y + 1/z >= 9.
LHS >= 3/cuberoot(xyz) >= 3/((x+y+z)/3) = 9.
