Suppose $f$ is a Lebesgue integrable function on $X$ and define:
\begin{align*} X_{n}=\{x \in X: |f(x)|\geq n\} \end{align*}
Show that $\lim_{n \to \infty} n\lambda(X_{n})=0$
Proof: Since $f$ is integrable, then $|f|$ is also integrable. Consider the following inequality:
$$|f|\chi_{X_{n}} \leq |f|$$ Using properties of Lebesgue integral then:
$$\int |f|\chi_{X_{n}} d\lambda \leq \int |f| d\lambda < \infty$$ So
$$0 \leq\int n\chi_{X_{n}}d\lambda=n\lambda(X_{n})\leq\int|f|\chi_{X_{n}}d\lambda<\infty$$
Taking limit as $n \to \infty$ and using Lebesgue increasing convergence:
$$0 \leq \lim_{n \to \infty} n\lambda(X_{n}) \leq \int |f| \lim_{n \to \infty} \chi_{X_{n}} \to 0$$
Is the proof correct?
The last step is not correct. The sequence is not increasing; it is decreasing. It is better to finish the proof using DCT.