I'm trying to show that if $X_n$ converges in probability to 0 and $Y_n$ converges in probability to 0, then $X_n+Y_n$ converges in probability to $0$, ie the sum rule for probability limits.
What I'm trying to do is to show it directly using $\epsilon, \delta$ arguments as opposed to appealing to the CMT, to improve my understanding of convergence. But I get stuck almost immediately because I don't know how I can use bounds for the probability each sequence exceeds some value to get a bound for the probability the sum exceeds some value...
Here's where I get stuck:
For every $\epsilon>0$ and every $\delta>0$ I have an $N$ s.t $n>N \implies P( |X_n|>\epsilon)<\delta$ and $P(|Y_n|>\epsilon)<\delta$. I also know that $|X_n+Y_n| \leq |X_n|+|Y_n|$ and that therefore $P( |X_n+Y_n|>\epsilon) \leq P( |X_n|+|Y_n| >\epsilon)$. Now it seems obvious that if $P( |X_n|>\epsilon)$ and $P(|Y_n|>\epsilon)$ can be made as small as we want, the RHS can be bounded, but I can't quite work out how. I can't seem to relate the probability of the sum to the probabilities I can bound.