Prove the convergence in distribution of random variables Let $U \in \mathbb{R}^d$ be a discrete valued random variable and $Z \in \mathbb{R}^d$ be continuous random variable. Both $U$ and $Z$ have finite mean and variance and are independent. I want to prove the following statements as $\varepsilon \to 0$:
\begin{align}
(a)\ \  &U+ \varepsilon Z  \xrightarrow{\mathcal{d}} U \\
(b) \ \ & (U,U+\varepsilon Z) \xrightarrow{\mathcal{d}} (U,U) 
\end{align}
where $X_n \xrightarrow{d} Y$ means $X_n$ converges in law to $Y$.
Can anyone guide me in this?
My attempt: Let $\Phi$ denote the characteristic function. So $\Phi_{U+\varepsilon Z}(t)=\Phi_U(t) \Phi_{\varepsilon Z}(t)=\Phi_U(t) \Phi_Z(t\varepsilon)=\Phi_U(t) \mathbb{E}[e^{i t \varepsilon Z}]$. Since $|e^{i t \varepsilon Z}|=1$ and $e^{i t \varepsilon Z} \to 1$ as $\varepsilon \to 0$, is it justified to say that $\mathbb{E}[e^{i t \varepsilon Z}] \to 1$ and thus $ U+ \varepsilon Z  \xrightarrow{\mathcal{d}} U $?
I think the same reasoning should also hold for $(b)$.
 A: Let me suggest a proof that does not use the characteristic function.
We say that $X_n \xrightarrow[]{d} X$ when (See the Portemanteau Theorem)
  $$ \text{For every open set } A: \liminf_n\Bbb{P}(X_n \in A)\geq \Bbb{P}(X \in A)$$
or equivalently
  $$\text{For every closed set } F:\limsup_n\Bbb{P}(X_n \in F)\leq \Bbb{P}(X \in F)$$  
In the case a), define $F^\delta = \{x \mid d(x,F)<\delta\}$
$$\Bbb{P}(U + \epsilon Z \in F )  \leq \Bbb{P} (U \in F^\delta) + \Bbb{P} (|\epsilon Z| \geq \delta) $$
Therefore for every $\delta >0$
 $$\limsup_{\epsilon \to 0}\Bbb{P}(U + \epsilon Z \in F )  \leq \Bbb{P} (U \in F^\delta) $$
This implies that
 $$\limsup_{\epsilon \to 0}\Bbb{P}(U + \epsilon Z \in F )  \leq \Bbb{P} (U \in F) $$
so $U + \epsilon Z\xrightarrow[\epsilon \to 0]{d} U $
For the case b)
consider the closed set $F$
 $$\Bbb{P}( (U,U + \epsilon Z) \in F )  \leq \Bbb{P} ((U,U) \in F^\delta) + \Bbb{P} (|\epsilon Z| \geq \delta) $$
Therefore for every $\delta >0$
 $$\limsup_{\epsilon \to 0}\Bbb{P}( (U,U + \epsilon Z) \in F)  \leq \Bbb{P} ((U,U) \in F^\delta) $$
This implies that
 $$\limsup_{\epsilon \to 0}\Bbb{P}( (U,U + \epsilon Z) \in F)  \leq \Bbb{P} ((U,U) \in F) $$
so $(U,U + \epsilon Z)\xrightarrow[\epsilon \to 0]{d} (U,U) $
