Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 28 / 6 / a / Solution

For $\epsilon >0$, the triangle inequality and the union bound giveThus $\boxed{X_n+Y_n⟶X+Y\text{ in probability}}$. This proves closure of convergence in probability under addition. No independence is needed; the random variables are compared on their given common probability space.