Suppose f_n:X->[0,inf] is a sequence of measurable functions with

f_1 >/= f_2 >/=...>/=0 and that f_n->f as n goes to infinity pointwise. and f_1 is in L^1

Prove that the limit integral of f_n = the integral of f.

Since f_1 is in L^1 can we say that f_2, f_3.... are in L^1 by Lebesgue's dominated convergence theorem? If not, I do not know what to do. If yes, then I have been trying to work with partial sums, but I haven't gotten anywhere.