Prove $ \lim_{n \to \infty} \int_a^b \left (1+\frac x n \right)^ne^{-x} dx = b-a $

Question

So I know if you let $f_n(x) = \left(1+\frac x n \right)^ne^{-x}$, then $f(x) = e^{x}\cdot e^{-x} = 1$

Thus, the integral from $\int_a^b dx$ = $b-a$. I'm confused about how we know $f_n$ is uniformly continuous.

Answer 1

Over any bounded subset, $(1+n^{-1}x)^n\to e^x$ uniformly.