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Let A1, A2, A3 and an be positive numbers, and prove the inequality (a1 + A2 +. + an) (1 / A1 + 1 / A2 +. + 1 / an) ≥ n & # 178;
It can be proved by Cauchy inequality,
Cauchy Inequality: (A1 & # 178; + A2 & # 178; +...) +an²)(b1²+b2²+… +bn²)≥(a1b1+a2b2+… +anbn)²
So there are (a1 + A2 +. + an) (1 / A1 + 1 / A2 +. + 1 / an) ≥ (√ 1 + √ 1 +...) +√1)=n²
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