Let $f_n : \mathbb{R}^+ \to \mathbb{R} : x \mapsto (1-\frac{x}{n})^n \mathbb{1}_{[0,n]}$. I would like to prove that the sequence $(f_n)_n$ converge uniformly on $\mathbb{R}^+$to $x \mapsto e^{-x}$
Normally there is general strategy of studying the maximum of : $x \mapsto e^{-x} - f_n(x)$ and then prove that this maximum goes to $0$ as $n \to \infty$. Yet I am completely stuck in findind this maximum.
First taking the derivative I get : $\forall x \in [0,n], -e^{-x} + (1-\frac{x}{n})^{n-1}$ so the derivative has kind of the same of form so it doesn't really help in finding the maximum. Maybe there is an other way to prove the uniform convergence without studying the maximum, but then I don't really know.
Thank you !
For $t \geq 0$ we have $e^{-t} \geq 1-t$. Hence $e^{-x} =(e^{-x/n})^{n} \geq (1-\frac x n)^{n}$. It follows that $|(1-\frac x n)^{n}-e^{-x}| \leq 2e^{-x} <\epsilon $ whenever $x >\log (2/\epsilon)$. For uniform convergence on $[0, \log (2/\epsilon)]$ apply Dini's Theorem. [You can show that $e^{-x}-(1-\frac x n)^{n}$ is a decreasing sequence of continuous functions converging pointwise to $0$].