I was going through the proof for the limit representation of the Gamma function where to prove the interchange of the limit and integral is justified, the author uses the following two relationships to show $$\bf{0<e^{-t} - (1-\frac{t}{n})^n<\frac{t^2}{2n}, (0<t<n)}$$:
(i) $$\bf{1-e^{t}\left(1-\frac{t}{n}\right)^n = \int_0^t e^{\tau}\left(1-\frac{\tau}{n}\right)^n\frac{\tau}{n}d\tau}$$
(ii) $$\bf{0 < \int_0^t e^{\tau}\left(1-\frac{\tau}{n}\right)^n\frac{\tau}{n}d\tau < \int_0^t e^{\tau}\frac{\tau}{n}d\tau = e^t\frac{t^2}{2n}}$$
Even though differentiating both sides of (i) is giving me LHS = RHS, I am not able to prove or understand where this relation is coming from. Also unable to prove the equality in (ii). Please help.
If so if $0 < \tau <t $ then $ O <\frac{ \tau}{n} < 1$ so clue is prove $0< x(1-x)<x$ wich after division by non zero x gives $1-x < 1$ wich is true for. $0< x< 1 $ becaulse such relations we have in integrals