Poitwise limit of a monotonically increasing function sequence in $L^p(M)$ exists always almost everywhere

Question

I am having trouble comprehending the following remark made in my function analysis text.

Let $\left(f_n\right)_{n\in \mathbb{N}}$ be a sequence of functions in $L^p(M), 1 \leq p < \infty$ and assume that $0 \leq f_{n} \leq f_{n + 1}$ $\mu$-a.e. in $M$, $n \in \mathbb{N}$. Then the pointwise limit $f = \lim_{n\to \infty}f_n$ exists $\mu$-a.e. in $M$.

What I don't understand is that what exactly guarantees the existence? The function sequence $f_n = n\cdot 1_{[0, 1]}$ in e.g. $L^p(\mathbb{R})$ satisfies the criterions but $\lim_{n\to\infty}f_n := \infty \cdot 1_{[0, 1]}$ seems a bit dubious.