Let $f\in L^1(\mathbb{R^n,R})$ be a non-negative function. Show that if $\int f=0$ then $f=0$ almost everywhere. Hint: Consider the sequence $f_k = kf$.
I've seen that there are lots of proofs of this theorem, my problem is that we are not allowed to use definitions and theorems of measure-theory (I mean the definition of a general measure $\mu$), because we "have not learned it yet". We've seen theorems like
Monotone convergence theorem: Let($f_k)_{k\in\mathbb{N}}$ ⊂ $L^1(\mathbb{R^n,R})$ be a monotone increasing sequence such that $( f_k)_{k\in\mathbb{N}}$ is bounded. Then $f_k$ converges a.e to a function $f\in L^1(\mathbb{R^n, R})$ and $\int f=\lim_{k\rightarrow\infty}\int f_k$.
I think I should use this theorem but it gets quite consfusiong..
Let $E=\{x:f(x)>0\}$. Then $0\leq\int_E f\leq \int_X f=0$ so $\int_E f=0$. Let $f_k(x)=kf(x)$. Then $f_k$ is monotone increasing and $f_k \to g$ where $$ g(x)=\begin{cases}0,&x\not\in E\\ +\infty,&x \in E \end{cases} $$ By monotone convergence theorem, how can you finish it?