Let $(x_n)$ be any function sequence such that
$ \int_0^1x_n(t)dt=1 \qquad \forall n $
$ \lim_{n\to\infty}x_n = x $
I'm trying to prove that the limit $x$ also has the property $\int_0^1x(t)dt=1$. I don't think I could construct a "bounding" function to use the dominated convergence theorem. Could I have a hint?