- If $X_n\overset{p}{\rightarrow}X,Y_n\overset{p}{\rightarrow}Y$ then, show that $X_n+Y_n\overset{p}{\rightarrow}X+Y$.
I am not looking for the proof. My issue is, this seems contradictory to Slutsky's theorem, which is as follows.
- If $X_n\overset{p}{\rightarrow}X,Y_n\overset{p}{\rightarrow}c\in\mathbb{R}$ then, $X_n+Y_n\overset{p}{\rightarrow}X+c$. [Replacing all convergences in distribution by convergence in probability as mentioned on Wikipedia]. Also, for $Y_n\overset{p}{\rightarrow}c$, $c$ being a constant is a necessity otherwise, the result is not guaranteed.
However, if theorem 1 holds then this necessity is useless. Slutsky's theorem should be true regardless of $c$ being a constant or a random variable. Where am I wrong? Is it because the necessity is only valid if we only have $X_n\overset{d}{\rightarrow}X$?