# Is the converse of this convergence theorem about characteristic functions true?

In Kai Lai Chung's A Course in Probability Theory,

1. Theorem 6.3.1 says:

2. Then he provides Theorem 6.3.2, which is not included in this post, and then gives a corollary of Theorem 6.3.2,

I was wondering if the above corollary of Theorem 6.3.2 is the reverse of Theorem 6.3.1, under the same condition of Theorem 6.3.1: $\{\mu_n, 1\leq n \leq \infty\}$ be p.m.'s on $\mathbb R^1$ with ch.f.'s $\{f_n, 1 \leq n \leq \infty\}$?

In other words, is the condition "$\{\mu_n, 1\leq n \leq \infty\}$ be p.m.'s on $\mathbb R^1$ with ch.f.'s $\{f_n, 1 \leq n \leq \infty\}$" in Theorem 6.3.1 same as the condition " $\{\mu_n, 1\leq n \leq \infty\}$ and ch.f.'s $\{f_n, 1 \leq n \leq \infty\}$ are corresponding p.m.'s and ch.f.'s" in the corolary of Theorem 6.3.2?

Thanks and regards!

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