I have found the following very useful theorem and I would appreciate some help comprehending it fully.
Theorem Let $\{X_n \} $ be a sequence of random variables bounded in probability and let $ \{Y_n \} $ be a sequence of RVs which converge to $0$ in probability. Then
$$X_n Y_n \rightarrow 0 \quad \text{in probability} $$
And its proof:
For a given $\epsilon$ since $X_n$ is bounded in probability we can choose $N$ and a constant $B$ such that
$$n \geq N \Rightarrow P \left[ |X_n| \leq B \right] \geq 1-\epsilon$$
Then
$$ \overline{\lim_{n \to \infty}}P \left[|X_n Y_n| \geq \epsilon \right] \leq \overline{\lim_{n \to \infty}} P \left[ |X_n Y_n| \geq \epsilon, |X_n| \leq B \right]+ \overline{\lim_{n \to \infty}}P \left[|X_n Y_n| \geq \epsilon, |X_n| > B \right] \leq \overline{\lim_{n \to \infty}} P\left[|Y_n| \geq \epsilon/B \right] +\epsilon=\epsilon $$
If someone could help me understand how the last inequality is derived, given the information we have, I would be grateful. I understand the second to last inequality comes from the subadditivity property of limitsuperior so I merely need to comprehend the last one. Thank you in advance.
EDIT: I think it is because $$P(A) \times P(B|A) \leq P(A) $$ and the complement of the event that is implied by the bound in probability.