Does the central limit theorem imply the law of large numbers

Question

Assuming that the distribution has finite variance (a condition not required for the LLN), then doesn't the LLN follow from the CLT?

Answer 1

WLLN, yes.

Here is a general claim: Suppose $\{ f_n \}$, $f$, and $g$ are random variables, and

$$ \sqrt{n} (f_n - f) \stackrel{d}{\mapsto} g. $$

Let's say the CDF of $g$ is continuous everywhere. Then $f_n \rightarrow f$ in probability. This is because $\sqrt{n} (f_n - f)$ is bounded in probability/uniformly tight.