Define the following integral with $n$ an integer greater than $1$:
$$I_{n}=\int_{0}^{1}\frac{e^t}{(1+t)^n}dt.$$
Is it true that for all $n \geq 2$,
$$ \frac{1}{n-1}\left(1-\frac{1}{2^{n-1}}\right)\leq I_{n} \leq \frac{e}{n-1}\left(1-\frac{1}{2^{n-1}}\right)?$$