Let $(X,\mu,\mathcal{A})$ be a complete, $\sigma$-finite measure space. Let $(f_j)\in\mathcal{L}_1(X,\mu,\mathbb{R})^\mathbb{N}$ be an increasing sequence with $f_j\geq 0$ and $f_j\to f$ $\mu$-a.e. for some $f\in\mathcal{L}_1(X,\mu,\mathbb{R})$.
I want to show that $(f_j)$ is a $\mathcal{L}_1$-cauchy sequence. How? I don't want to use the monotone convergence theorem, Fatou's lemma, or the DCT.
This should work: $(\Vert f_j\Vert_1)$ is an increasing sequence in $\mathbb{R}$, bounded by $\Vert f\Vert_1$. So, it is convergent and, in particular, a cauchy sequence.
So, given $\varepsilon>0$, we have some $N\in\mathbb{N}$ such that we have $\vert \Vert f_j\Vert_1-\Vert f_k\Vert_1\vert<\varepsilon$ for all $j,k\geq N$. So for $j,k\geq N$ and wlog $j\geq k$ this means $$\begin{align*}\Vert f_j-f_k\Vert_1 &= \int \vert f_j-f_k\vert d\mu\\ &=\int (f_j-f_k)d\mu\\ &=\int f_jd\mu - \int f_kd\mu\\ & = \int \vert f_j\vert d\mu-\int \vert f_k\vert d\mu\\ &=\left\vert\Vert f_j\Vert_1 -\Vert f_k\Vert_1\right\vert<\varepsilon,\end{align*}$$
that is $(f_j)$ is a $\mathcal{L}_1$-cauchy sequence.