<s>[INST] Proof $(1+1/n)^n$ is an increasing sequence I need help proving $a_n=\left(\dfrac{n+1}{n}\right)^n$ is increasing sequence on the positive integers.
An exercise in the analysis book by Mattuck asks to prove $a_n=\left(\dfrac{2^n+1}{2^n}\right)^{2^n}$ is increasing.
But this is easy since $\left(\dfrac{2^n+1}{2^n}\right)^{2^n}=\left(\left(\dfrac{2^n+1}{2^n}\right)^{2}\right)^{2^{n-1}} =\left(\dfrac{2^{2n}+2\left(2^n\right)+1}{2^{2n}}\right)^{2^{n-1}}=$
$\left(\dfrac{2^{n-1}+1+\frac{1}{2^{n+1}}}{2^{n-1}}\right)^{2^{n-1}}>\left(\dfrac{2^{n-1}+1}{2^{n-1}}\right)^{2^{n-1}}$ as desired.
I am having difficulty in the general case:
I must prove $\left(\frac{n+2}{n+1}\right)^{n+1}>\left(\frac{n+1}{n}\right)^n$.
This is equivalent to $(n+2)^{n+1}n^n>(n+1)^{2n+1}$ which is the same as proving $\frac{n+2}{n+1}>\left(\frac{(n+2)(n)}{(n+1)^2}\right)^n$ which is the same as $\frac{n+2}{n+1}>\left(1-\frac{1}{(n+1)^2}\right)^n$
I'm stumped. Thanks in advance. Oh, and I don't want to use the derivative.
 [/INST] Use the Bernoulli inequality,
$$1+(n+1)(a-1)\leq a^{n+1}$$ with 
$$a=\frac{n(n+1)}{(n+1)^2}$$
</s>