I wanted to prove $\lim_{n\to\infty} \int_0^\infty (1-x/n)^n \, dx$ converges using Lebesgue integrable techniques. (I wolfram'd this to see that this converges to $+1$).
What I have tried so far:
I tried to use DCT but ran into a problem: I know that it is possible to bound $(1-x/n)^n$ for all $x\in[0,\infty)$, however bounding this by an integrable function on $x\geq n$ is quite hard, I don't think it is possible.
My next thought was to define $f_n(x) := (1-x/n)^n$ if $x<n$ and $f_n(x) = 0$ otherwise. I know these $f_n$'s converge to the same function as the sequence $\{(1-x/n)^n\}$ but I'm not quite sure how I would show that, $$\lim_{n\to\infty} \int_0^n (1-x/n)^n \, dx = \lim_{n\to\infty} \int_0^\infty (1-x/n)^n \, dx$$ Any help is greatly appreciated!