Convergence in probability Questions We want to show that if ${X_n}$ and ${Y_n}$ are two sequences of random variables that converge in probability to ${c_1}$ and ${c_2}$, then  ${X_n+Y_n}$ converges in probability to $c_1+c_2$, and ${X_nY_n}$ converges to $c_1c_2$.
Approach: I was thinking of using Chebyshev's Inequality but that was not working and I was having difficulty using a standard epsilon delta proof. How do I do this problem?
 A: We are given that $P(|X_n-c_1|\geq \epsilon)\to0$ and $P(|Y_n-c_2|\geq \epsilon)\to0$.
We know that $$P(|X_n-c_1|\geq\epsilon\ AND\ |Y_n-c_2|\geq\epsilon)\leq P(|X_n-c_1|\geq \epsilon)+P(|Y_n-c_2|\geq \epsilon)$$
and that 
$$P(|X_n+Y_n-c_1-c_2|\geq2\epsilon)\leq P(|X_n-c_1|\geq\epsilon\ AND\ |Y_n-c_2|\geq\epsilon)$$
Therefore $$P(|X_n+Y_n-c_1-c_2|\geq2\epsilon)\to0$$
The second inequality comes from the inclusion of the two sets. The inclusion follows from the triangle inequality $$|X_n-c_1+Y_n-c_2|\leq|X_n-c_1|+|Y_n-c_2|.$$

The one for the product is slightly more delicate. We need to use the triangle inequality together with the following trick.
$$|X_nY_n-c_1c_2|=|X_nY_n-X_nc_2+X_nc_2-c_1c_2|\leq|X_n||Y_n-c_2|+|c_2||X_n-c_1|$$
The next problem to overcome comes from that $|X_n|$ in the first term. We need to use that 
$$|X_n|\leq|X_n-c_1|+|c_1|$$
A: Hint: Slutsky's Theorem
This allows you to treat arithmetic with random variables like arithmetic in algebra under certain convergence conditions.
