How to solve $\left(1+\frac{1}{n+2}\right)^{n+2}\geq \left(1+\frac{1}{n+1}\right)^{n+1}$ for all $n\in \mathbb{N}$

Question

any impulses, suggestions? I have been trying for a while but it doesn't get me anywhere...

Kind regards

Answer 1

Consider the function $$f(x)=(1+\frac{1}{x})^{x}$$.

Now try and show that $f'(x)>0$ for all $x>0$

Answer 2

Hint: Prove that $f(x)=(1+\frac{1}{x})^x$ is strictly increasing for x>0

Answer 3

log both sides and rearrange:sufficient to prove $$\frac{n+2}{n+1}\ge\ln(\frac{\frac{n+2}{n+1}}{\frac{n+3}{n+2}})$$ which is $$\ln(e^{\frac{n+2}{n+1}})\ge\ln(1+\frac{1}{(n+1)(n+3)})$$ exponential both sides and LHS$>2>$RHS