Prove $n \mu(|f| \geq n) \to 0$ for any integrable $f$

Question

Let $f$ be integrable; $$\int |f| d\mu < \infty$$ Prove that: $$\lim_{n \to \infty} n \mu (E_n)= 0$$ where $$E_n = \{x: |f(x)| \ge n \}$$

Intuitively it is quite obvious. I don't know how to write a formal proof however.
I tried to create a sequence of functions $f_n$ which would be increasing and then use one of Lebesgue's theorems.

Answer 1

Hint:

$$n \mu(E_n) = \int_{|f(x)| \geq n} n \, \mu(dx) \leq \int_{|f(x)| \geq n} |f(x)| \, \mu(dx).$$