I'm stuck trying to find an example of a sequence of functions $\{f_n\}_{n=1}^{\infty}$, which is nonnegative, such that $f_n \rightarrow 0$ uniformly and $\int\limits_0^\infty f_n=1$.
My first thought was a series of triangles of area 1 that move away from the origin that get taller and more narrow as they get further away.