# About a Lebesgue intergrability condition

Let $$X$$ be a random variable in the probability space $$(\Omega, F, P)$$ such that for all $$\epsilon > 0$$ there exists a $$c > 0$$ such that $$\int\limits_{|X| > c} |X| dP < \epsilon$$ Prove that $$X$$ is integrable.

I attempted to use the dominated converange theorem, which states like this in my textbook

Let $$Y, X_1, X_2,...$$ be random variables such that $$|X_n| < Y$$ for all $$n$$ and $$EY < \infty$$. If $$X_n \to X$$ then $$X$$ is integrable.

but building a sequence $$X_n$$ dominated by some random integrable variable that converges to $$X$$ is something I don't have any insight to do.

Please give me a hint. Anything is greatly appreciated. Thank you.

Hint: $$\left\lvert X\right\rvert=\left\lvert X\right\rvert\mathbf 1\left\{\left\lvert X\right\rvert\leqslant c\right\}+\left\lvert X\right\rvert\mathbf 1\left\{\left\lvert X\right\rvert\gt c\right\}\leqslant c+\left\lvert X\right\rvert\mathbf 1\left\{\left\lvert X\right\rvert\gt c\right\}.$$ Apply this to $$c$$ such that (for example) $$\int\limits_{|X| > c} |X| dP < 1$$.