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a question about the proof that $f \in L^1 \Rightarrow \lim_{x \to \infty} f(x) = 0$

if $f \in L^1$, prove that:
$\lim_{x \to \infty} f(x) = 0$
My book says
Since $f \in L^1$ then $\|f\|_1 = \int_{[a,\infty]}|f|dx \le \|f\|_\infty \cdot \int_{[a,\infty]}1 \to 0$ as $ \int_{[a,\infty]}1 \to 0$
is this right?

A:

You have:
$$
\|f\|_1 \leq \|f\|_\infty \cdot \int_{[a,\infty)}1 = \|f\|_\infty \to 0
$$
And to prove the converse:
$$
0\leq \|f\|_1 = \int_{[a,\infty)}|f|dx \leq \int_{[a,\infty)}|f|\cdot 1\to 0
$$

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