Let and let for . Prove

*Proof.*First,

for all . We then apply the Cauchy-Schwarz inequality to the numbers and . So, Cauchy-Schwarz gives us,

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Stumbling Robot

A Fraction of a Dot
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Prove the product of sums of n numbers and their reciprocals is greater than n^2

* Proof. * First,

Let and let for . Prove

for all . We then apply the Cauchy-Schwarz inequality to the numbers and . So, Cauchy-Schwarz gives us,