First you should show that: $$P[|X_n Y_n| \geq \epsilon,|X_n|\leq B]\leq P[|Y_n|\geq \epsilon/B]$$ Hint: If $A\subset B$ then $P(A)\leq P(B)$.
For the 2nd part, again by using above property, you can say that: $$P \left[|X_n Y_n| \geq \epsilon, |X_n| &gt; B \right]\leq P[|X_n|&gt;B]$$ Now try to use $P[|X_n|\leq B]\geq 1-\epsilon$.

Convergence in probability of a product of RVs
I'm using devise v.2.2.4 on my Rails 3.2.17 and I need some features related with security policies.

The admin user will change the security policies at any time like show next image:


but I don't know how make it with devise, because devise read configuration of initializers/devise.rb and on production all initializer are loading the first time only.
