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proof of Cauchy-Schwarz inequality

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i use cauchy-schwarz inequality to prove the follow inequality

Let (summation from i=1 to n) = S and X = multiply

S[a(suffix_i)]^(1/2) <or= {nXS[a(suffix_i)]}^(1/2)

it has no problem
the problem is "hence"

hence, prove that

1 + 1/2 + ... + 1/n <or= (2n-1)^(1/2)

i cannot do it

Are you sure that you quoted the original statement correctly? To me this does not look like something one would need C-S-inequality for as it is trivial.

"Do not meddle in the affairs of wizards for they are subtle and quick to anger."

Cases n=1 and n=2 are trivial to check. Now, sum 1/k <= sqrt( n sum 1/k^2) < sqrt(n pi^2/6) < sqrt (2n-1) as n>2. Summation index is k and it goes from 1 to n.

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