= Solution
For $\varepsilon>0$, the <triangle inequality> and the <union bound> give
$$
\begin{aligned}
\mathbb P\bigl(|(X_n+Y_n)-(X+Y)|>\varepsilon\bigr)
&\leq\mathbb P(|X_n-X|>\varepsilon/2)
+\mathbb P(|Y_n-Y|>\varepsilon/2)\\
&\longrightarrow0.
\end{aligned}
$$
Thus $\boxed{X_n+Y_n\longrightarrow X+Y\text{ in probability}}$. This proves closure of <convergence in probability> under addition. No <independence> is needed; the <random variables> are compared on their given common <probability space>.
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