Let $f$ a non-negative Lebesgue integrable function $ ( f\in L^+([0 ,\infty)) )$ such that $\displaystyle{ \lim_{x \to +\infty} f(x)}$ exists (finite or infinity). Prove that $\displaystyle{\lim_{x \to +\infty} f(x) =0}$.
Here it is the only thing I did.
Consider an increasing sequence $ (f)_n$ of simple non-negative functions such that $ f_n \to f$ pointwise. Then $ \int f =\lim \int f_n$.
1st case: $\displaystyle{\lim_{x \to +\infty} f(x) = l < +\infty}$.
Let $ \epsilon >0$, then there exists $ r>0 $ such that $ |f(x)-l|< \epsilon , \quad \forall x>r$.