<s>[INST] Can someone shed some light on this inequality? I have a question relating to image that I've attached. It is a proof that the sequence is increasing. I don't understand the logic behind the third equation $$\frac{a_{n+1}}{a_n}>\left (1-\frac{1}{n+1}\right ) \left (\frac{n+1}{n}\right)$$
where does the equation in the first and second parenthesis come from? 
Ok, I have another relating question:
why $$\frac{a_{n+1}}{a_n}> (1+\frac{1}{n})$$ ( The expression of third line.

 [/INST] So, we have
$$\frac{a_{n+1}}{a_n} = \left(1 - \frac{1}{(n+1)^2}\right)^{n+1}\left(\frac{n+1}{n}\right).$$
The author then applies Bernoulli's inequality to the first term on the RHS:
$$\left(1 - \frac{1}{(n+1)^2}\right)^{n+1} > 1 + (n+1)\left(\frac{-1}{(n+1)^2}\right) = 1 - \frac{1}{n+1}.$$
We can now return to the first equation and utilize this estimate; namely, we have
$$\frac{a_{n+1}}{a_n} = \left(1 - \frac{1}{(n+1)^2}\right)^{n+1}\left(\frac{n+1}{n}\right) > \left(1-\frac{1}{n+1}\right)\left(\frac{n+1}{n}\right).$$
Finally, we multiply out the RHS of the inequality 
$$\frac{a_{n+1}}{a_n} > \left(1-\frac{1}{n+1}\right)\left(\frac{n+1}{n}\right) = \frac{n+1}{n} - \frac{1}{n} = 1.$$
So, we have
$$\frac{a_{n+1}}{a_n} > 1 \implies a_{n+1} > a_n,$$
which means that $\{a_n\}$ is an increasing sequence. 
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