Can someone please show me how to prove this inequality: Let a > b > 0 and let n be a natural number greater than or equal to 2. Then a^(1/n) - b^(1/n) < (a-b)^(1/n). Thank you for your help. It is greatly appreciated.
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Originally Posted by jamesHADDY Can someone please show me how to prove this inequality: Let a > b > 0 and let n be a natural number greater than or equal to 2. Then a^(1/n) - b^(1/n) < (a-b)^(1/n). Thank you for your help. It is greatly appreciated. We can prove something stronger. Let be a real number. Let . Then thus for . This means . Now if . Using the above result it means . Thus, if then and the above result holds.
Thank you for answering. Is there a little bit easier way to show this (possibly without integration)? Maybe with sequences or mean value theorem?
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