Prove the following integral is asymptotically zero

Question

I have to solve the following exercise. I would appreciate to get a hint for it. Suppose $(\Omega, \mathcal{F}, \mu)$ be a measure space. Let $f$ be an integrable function. Show $$\lim_{n\to\infty}\int_{\{w:|f(w)|>n\}}|f|d\mu=0.$$

Answer 1

$$|f|1\{|f|>n\}\le |f|\in L^1.$$

Thus,

$$\lim_{n\to\infty}\int|f|1\{|f|>n\}d\mu=\int \lim_{n\to\infty}|f|1\{|f|>n\}d\mu.$$