Definition: Let $(X, \mathcal{A}, μ)$ be a measure space. Let $f:X\rightarrow \mathbb{R}$ be a measurable function. Let $E\in \mathcal{A}$. Then $$\int_E fdμ:=\int_X (f\chi_E)dμ.$$
Let $(X, \mathcal{A}, μ)$ be a measure space. Let $f:X\rightarrow \mathbb{R}$ be a measurable function. Let $E\in \mathcal{A}$. Assume that $\exists M>0$ s.t $$|f(x)|\le M, \forall x\in E.$$ Then prove that $$|\int_E f dμ| \le Mμ(E).$$
Please help me in this proof. Definition is given above the proof of the statement.
My try:
$$|\int_E f dμ|=|\int_X (f \chi_E) dμ|$$ $$ \le \int_X |(f \chi_E)| dμ$$ $$= \int_X |f| |\chi_E| dμ$$ $$ \le \int_X M |\chi_E| dμ $$ $$= M \int_X |\chi_E| dμ$$ $$= M μ(E) $$