convergence in law in $\mathbb{R}^d$

Question

In my probability professor's notes there's this lemma given as obvious, but I am not able to prove its truth.

Let $(X_n)_{n \in \mathbb{N}}$ and $X$ be random variables with values in $\mathbb{R}^d$.

Prove that the following to statements are equivalent:

1. the sequence $(X_n)_{n \in \mathbb{N}}$ converges in law to $X$.
2. for any $u \in \mathbb{R}^d$ we have that $(\langle u,X_n \rangle)_{n \in \mathbb{N}}$ converges in law to $\langle u,X \rangle$.

I am able to prove the easy direction (i.e. 1. $\implies$ 2.).

Can somebody provide me a proof of 2. $\implies$ 1. ?

Answer 1

This is the Cramér-Wold theorem. A proof is given in Billingsley's Probability and Measure (p.383 Theorem 29.4). This is a very standard reference.

The way in Billingsley to prove "2 implies 1" is using the characteristic functions (and the continuity theorem). A key observation is that for any $t\in{\bf R}^d$, $$ \varphi_X(t)= \varphi_{\langle t,X\rangle}(1) $$ where $\varphi_X$ denotes the characteristic function of the random vector $X$ and $\varphi_{\langle t,X\rangle}$ is the characteristic function of the random variable $\langle t,X\rangle$.