I was attempting this question, which asks to prove $$\frac{1}{2ne}<\frac{1}{e} - \left(1-\frac{1}{n}\right)^n<\frac{1}{ne}\qquad ,\forall n>1 \text{ and } n\in Z^+$$
I have the solution to this problem in the back of the book, but I don't want to look at the solution. I want to know how you would have approached the problem.
I tried to prove this by mathematical induction, but I was stuck here. My approach was
For $n = 2$
$$\frac{1}{e} - \left(1-\frac{1}{2}\right)^2 = \frac{1}{e} - \left(1 - 1 +\frac{1}{4}\right) = \frac{4-e}{4e}$$ Clearly, $$\frac{1}{2\times 2e}<\frac{4-e}{4e}<\frac{1}{2e} = \frac{2}{4e}$$ Assumed that the statement holds for $n$
For $n+1$, I was stuck here and couldn't simplify further, $$\frac{1}{e} - \left(\frac{n}{n+1}\right)^n\left(\frac{n}{n+1}\right)$$
I would like to know how you would have approached the problem and the multiple approaches you might think would work here.
