If a random variable converge in distribution then scalar multiplication also converges in distribution. Let c be a constant and suppose that $X_n\xrightarrow{d}X$, prove that
i)  $cX_n\xrightarrow{d}cX$ 
ii) $X_n+c\xrightarrow{d}X+c$
I've tried to use that:
$lim_{n\rightarrow\infty} F_{cX_n}(x) = lim_{n\rightarrow\infty} P(cX_n\leq x) = lim_{n\rightarrow\infty} P\left(X_n\leq \dfrac{x}{c}\right) = lim_{n\rightarrow\infty} F_{X_n}\left(\dfrac{x}{c}\right) $
But that's not exactly what I was looking for.
The sequence $X_n$ converges in distribution to $X$ as $n \rightarrow \infty$ (denoted $X_n \Rightarrow X$) if and only if, for any bounded continuous function $f$ (denoted $f \in C_b$), we have
$$\lim_{n\rightarrow\infty} E[f(X_n)] = E[f(X)].$$
Let $c \in \mathbb{R}$ be given. The maps $g_c^\times(x) = cx$ and $g_c^+(x) = x + c$ are continuous, so for any $f \in C_b$, $f\circ g_c^\times$ and $f\circ g_c^+$ are also in $C_b$. Therefore, because $X_n \Rightarrow X$ we have
$$\lim_{n\rightarrow\infty} E[f(X_n^{c,\times})] = \lim_{n\rightarrow\infty} E[(f\circ g_c^\times)(X_n)] = E[(f\circ g_c^\times)(X)] = E[f(X_c^\times)],$$
and ditto for $X_n^{c,+}$.
Remember that $X_n \xrightarrow{d} X$ it's by definition:
$\lim\limits_{n \rightarrow \infty} F_{X_n} (x) = F_X(x)$
Hence:
a)
\begin{equation*}
\begin{split}
\lim\limits_{n \rightarrow \infty} F_{cX_n} (x) &amp; = \lim\limits_{n \rightarrow \infty} \mathbb{P} (cX_n \leq x)  \\
 &amp; = \lim\limits_{n \rightarrow \infty} \mathbb{P} \left( X_n \leq \dfrac{x}{c} \right) \\
 &amp; = \lim\limits_{n \rightarrow \infty} F_{X_n} \left(\dfrac{x}{c}\right) \\
 &amp; =  F_{X} \left(\dfrac{x}{c}\right) \\
 &amp; =  \mathbb{P} \left( X \leq \dfrac{x}{c} \right) \\
 &amp; =  \mathbb{P} \left( cX \leq x \right) \\
 &amp; =  F_{cX} \left({x}\right) \\
\end{split}
\end{equation*}
\begin{equation*}
\therefore \lim\limits_{n \rightarrow \infty} F_{cX_n} (x) =  F_{cX} \left({x}\right) \end{equation*}
\begin{equation*}
\therefore cX_n \xrightarrow{d} cX
\end{equation*}
Analogously:
b)
\begin{equation*} 
\begin{split}
\lim\limits_{n \rightarrow \infty} F_{X_{n}+c} (x) &amp; = \lim\limits_{n \rightarrow \infty} \mathbb{P} (X_n + c \leq x)  \\
 &amp; = \lim\limits_{n \rightarrow \infty} \mathbb{P} \left( X_n \leq x-c \right) \\
 &amp; = \lim\limits_{n \rightarrow \infty} F_{X_n} \left( x-c \right) \\
 &amp; =  F_{X} \left( x-c \right) \\
 &amp; =  \mathbb{P} \left( X \leq x-c \right) \\
 &amp; =  \mathbb{P} \left( X+c \leq x \right) \\
  &amp; =  F_{X+c} \left( x \right) \\
\end{split}
\end{equation*}
\begin{equation*}
\therefore \lim\limits_{n \rightarrow \infty} F_{X_{n}+c} (x) =  F_{X+c} \left({x}\right)
\end{equation*}
\begin{equation*}
\therefore X_n+c \xrightarrow{d} X+c
\end{equation*}