Let $(f_n)_{n \in \mathbb{N}}$, $(g_n)_{n \in \mathbb{N}}$ be $L^1$-Cauchy-Sequences of step functions $X \rightarrow \mathbb{R}$ which converge to the same limit function almost everywhere.
Define $h_n = f_n - g_n$. Then $(h_n)_{n \in \mathbb{N}}$ is an $L^1$-Cauchy-Sequence which converges to zero almost everywhere (since the sum of Cauchy-Sequences itself is a Cauchy-Sequence).
The lecture notes now say
Since $|\int_X h \,d\mu| \leq ||h||_1$ we have $\int_X h_n \,d\mu \stackrel{n \rightarrow \infty}{\longrightarrow} 0$.
I don't see where this is coming from. It has to be simple, since no further steps are given. What am I missing?
In particular I don't see why the rhs of the inequality should converge to zero.