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I just can't get my head wrapped around a certain step in a proof that the sequence that approaches $e$ is monotonically increasing. The proof starts as follows:

We want to show that: $x_{n+1} \geq x_n \iff \\ \big(1 + \frac{1}{n+1}\big)^{n+1} > \big(1 + \frac{1}{n}\big)^{n} \iff \\ \big(\frac{(n+2)n}{(n+1)^2}\big)^n > \frac{n+1}{n+2} \iff ... \\ $

Now I have trouble understanding how we get from line 2 to line 3, anyone have a simple hint that can help me?

This question is not a duplicate because it asks for one certain step in an approach that is not covered in the linked question.

psyph
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  • I would just give a simple proof to this using AM-GM consider $n+1$ numbers one (1) and $(1+ \frac {1}{n} ) $for $n$ times and use AM - GM for instant happiness – 28ADY0901 Aug 15 '19 at 09:11

1 Answers1

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Multiply both sides of line 2 by $\frac {n^{n}} {(n+2) (n+1)^{n-1}}$ and simplify.

Kavi Rama Murthy
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