How do I show that this inequality holds for all $x \in (0, 1)$ and $n \in \Bbb N$?
$$ (1-x)^n<\frac{1}{1+nx}.$$
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2Possible duplicate of Proving with induction $(1-x)^n<\frac 1 {1+nx}$ – Martin R Jan 10 '19 at 18:35
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1Also: https://math.stackexchange.com/q/2078342/42969, https://math.stackexchange.com/q/1590586/42969 – Martin R Jan 10 '19 at 18:35
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Put $(1+x)^{-n}$ in between. – Aphelli Jan 10 '19 at 18:35
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Do you know Bernoulli's inequality ? – ViktorStein Jan 10 '19 at 18:36