Slutsky's Theorem

In the setting of Convergence of Probability Measures and Random Elements: Standing Notation, let $(X_{n})_{n\in\mathbb{N}}$ and $(Y_{n})_{n\in\mathbb{N}}$ be sequences of random variables on $(\Omega,\mathcal{F},P)$, let $X$ be a random variable on a probability space $(\Omega',\mathcal{F}',P')$, and let $c\in\mathbb{R}$. Suppose that $X_{n}\xrightarrow{d}X$ and that $Y_{n}\xrightarrow{P}c$ in the separable metric space $(\mathbb{R},d_{\mathbb{R}})$ (The Real Line and Euclidean Space Are Separable (i)).

(i) Sums. $X_{n}+Y_{n}\xrightarrow{d}X+c$.

(ii) Products. $Y_{n}X_{n}\xrightarrow{d}cX$.