Let n, m be natural numbers, and assume n < m. Prove that (m-n)/m > 1/(n+1) + 1/(n+2) + ... + 1/m.

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1/(n+1)+1/(n+2)+...+1/m < 1/n+1/n+...+1/n =(m-n)/n

because the sum has (m-n) terms

Hence

(m-n)/n > the given sum

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