Let $f\geqslant 0$. (i) Show that $\int f\wedge n\ \mathrm d\mu\uparrow \int f\ \mathrm d\mu $ as $n\to\infty$. (ii) Use (i) to conclude that if $g$ is integrable and $\varepsilon>0$ then we can pick $\delta>0$ so that $\mu(A)<\delta$ implies $\int_A |g|\ \mathrm d\mu <\varepsilon$.
In the above problem part - (i), I have the idea to define $f_n = \min(f, n)$ and then use the monotone convergence theorem. But this can be done only if we can show that $f_n$ is non-decreasing and $\lim_{n\rightarrow \infty} f_n = f$. I have already proved that the function is non-decreasing (by breaking it into cases and observing value of $f_{n+1} - f_{n}$), but I need some help with the second part.