Comparing sequences of random variables.

Question

Let $X_n$ be a a sequence of random variables such that $X_n > 0$ for all $n$ and $X_n \to_P 0$ as $n \to \infty$. Let $Y_n$ be another sequence of random variables such that $Y_n > 0$ and $Y_n \leq X_n$ with probability $p_n$.

If $p_n \to 1$, do we have $Y_n \to_P 0$?

This seems obvious, but I'm unable to come up with a rigorous argument for this.

Answer 1

Hint: $P(Y_n > \epsilon) = P(X_n \ge Y_n, Y_n > \epsilon) + P(X_n < Y_n, Y_n > \epsilon) \le P(X_n > \epsilon) + P(X_n < Y_n)$

Answer 2

Hint:

$$P(Y_n>\epsilon) \leq P\big(\{Y_n>\epsilon \} \cap \{Y_n \leq X_n\}\big) + P\big(\{Y_n>\epsilon \} \cap \{Y_n > X_n\}\big) $$$$ \leq P(X_n > \epsilon)+(1-p_n) \to 0$$

Can you fill in the justifications of each inequality?