Finiteness of expected values for independent random variables

Show that if the random variables X and Y are independent and for some $p > 0$: $E(|X+Y|^p)<\infty$, then $E(|X|^p)<\infty$ and $E(|Y|^p)<\infty$. I'd share thoughts or working out, but unfortunately I have none worth sharing.

Prove that for every $t> 0$ and $\epsilon > 0$, one has $P(|X+Y| > t) \ge P(|X| > t + \epsilon) P(|Y| \le \epsilon)$.

Then use the formula which gives absolute moments of a random variable $Z$ in terms of an integral of the distribution function $P(|Z| > t)$ : that is $E|Z|^p = \int_0^{\infty} p t^{p-1} P(|Z| >t)dt$.

EDIT : Further hints

For the first part, you just have to show that $\{|X| > t+ \epsilon, |Y| \le \epsilon \} \subset \{|X+Y| > t\}$ and then use independance.

For the second part, use the recalled formula to prove that $E|X|^p$ is finite, using the majoration $P(|X|>t) \le \frac{P(|X+Y| > t- \epsilon)}{P(|Y| \le \epsilon)}$, valid for all $t > \epsilon$.

(you can choose once for all $\epsilon > 0$ such that $P(|Y| \le \epsilon) > 0$ (why ?))