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I need to prove or disprove this inequality. So I am not too sure if the inequality even holds, Intuitively it does.

Prove or Disprove: $\dfrac{1}{n} < \dfrac{n-1}{n^2 - 2n}$ for any natural number > $3$.

My Work:

Base Case: $n=4$

$\dfrac{1}{4} < \dfrac{3}{8}$

Inductive Step: $n+1$

$\dfrac{1}{n+1} < \dfrac{(n+1)-1}{(n+1)^2 -2(n+1)} \implies \dfrac{n}{n^2 +2n +2-2n-1} \implies \dfrac{n}{n^2 + 1}$

Im a little stuck on what to do next. Any push in the right direction would help. Thanks!

hydra-bslash-0
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1 Answers1

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Applying induction is unnecessary. It is enough that you multiply both members of the inequality by $ n $ to arrive at what is equivalent to the inequality $ \frac {n-1}{n-2}> 1 $ which is true because $ n-1> n-2 $ trivially

bravoralph
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