# Proof of Levy's theorem and tightness

I am trying to understand the proof behind Levy's theorem. The statement is as follows

Theorem: Let $$(X_n)_{n\in \mathbb N}$$ be a family of random variables, $$(\mu_n)_{n\in \mathbb N}$$ their distributions and $$(\varphi_n)_{n\in \mathbb N}$$ their characteristic functions.

Suppose that for every $$t \in \mathbb R$$ $$\lim_{n\to \infty}\varphi_n(t)= \varphi(t)$$ and that $$\varphi$$ is continuous in $$0$$. Then there exists a distribution $$\mu$$ such that $$\varphi$$ is $$\mu$$'s characteristic function and $$(\mu_n)_{n \in \mathbb N}$$ converges weakly to $$\mu$$.

The proof that I have been provided with shows that the family $$(\mu_n)_{n\in \mathbb N}$$ is tight through an integral argument.

From my (limited) knowledge in probability, Prokhorov's theorem states that if the family $$(\mu_n)_{n\in \mathbb N}$$ is tight then there exists a distribution $$\mu$$ and a subsequence of the given family $$(\mu_{\kappa_n})_{n\in \mathbb N}$$ that converges weakly to $$\mu$$, but there is no general result about the convergence of $$(\mu_n)_{n \in \mathbb N}$$.

Supposing that we have shown that $$(\mu_n)_{n\in \mathbb N}$$ is tight, we can conclude that there exists a sequence $$\kappa_n$$ such that $$\lim_{n\to \infty}\varphi_{\kappa_n}(t)=\varphi(t)$$ but

How can we conclude that $$(\mu_n)_{n \in \mathbb N}$$ converges weakly to $$\mu$$?

1. Prokhorov's theorem actually says that $\textit{every}$ subsequence of your $(\mu_n)_{n \in \mathbb{N}}$ has a sub-subsequence converging in the weak topology to some probability measure.
2. By your condition on the sequence $(\phi_n)_{n \in \mathbb{N}}$ converging to $\phi$, every one of the sub-subsequences of $(\mu_n)_{n \in \mathbb{N}}$ must converge to the measure $\mu$ whose characteristic function is $\phi$.
That means limits are unique, and the sequence $(\mu_n)_{n \in \mathbb{N}}$ converges weakly to a probability measure $\mu$ if and only if each subsequence has a further subsequence converging weakly to that measure --- true because of steps 1, 2.