If a function is the limit in $L^1$ of simple functions, then that function is integrable

Question

I'm trying to prove that the completion of the integrable simple functions on $ (X,\Sigma,\mu) $ is $L^1(X,\Sigma,\mu)$. Suppose we have:

$lim_{n\to \infty}||f-s_n||_1=0$ for some sequence $(s_n)_n$ of simple integrable functions. How can I prove that $\int{|f(x)|}d\mu < \infty$?

Answer 1

Write $$\int |f| \ d\mu \leq \int |f - s_n| \ d\mu + \int |s_n| \ d\mu,$$ where $n$ is taken to be sufficiently large.