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I've been trying to solve this for hours, but I just can't seem to do it. And yes, I know I have to show that $(k+2)^{\frac{1}{k+2}} < ... < (k + 1)^{\frac{1}{k+1}}$ from the starting assumption that $(k+1)^{\frac{1}{k+1}} < k^{\frac{1}{k}}$. Any hints would be highly welcome.

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