Example of sequence with no dominating function but is uniformly integrable so limit holds

Question

Give an example of a sequence of functions on $\{f_n\}_{n=1}^\infty$ on $[0,\infty)$ such that $f_n\rightarrow 0$, the hypothesis of the Lebesgue dominated convergence theorem does not hold, but the sequence is uniformly integrable both locally and at infinity, and so $\lim_{n\rightarrow \infty} \int_{[0,\infty)} f_n(x)\,dx = 0$

Been trying to think of one for a week and a half... can't come up with anything... Would appreciate help on this one!! Also details on why it works is always appreciated!

Answer 1

Take $f_n=\frac 1 n I_{(n,n+1)}$. If $f_n \leq g$ for alll $n$ then $\int g=\infty$.