I have been asked to prove that $s/\sigma$ converges to 1 in probability, where $s$ and $\sigma$ are from a Normal distribution, $s$ representing the sample standard deviation and $\sigma$ is just our standard deviation.
I know from a previous example that I have solved that $s_n^2$ is a consistent estimator of $\sigma^2$. But now I don't know how to move ahead or what I could do from here.
Really hoping for some help on this .
Consider function $f(x) = \frac{\sqrt{x}}{\sigma}$ which is continious on $[0, \infty)$. We know that $s_n^2 \overset{P}{\to}\sigma^2$ hence $f(s_n^2) \overset{P}{\to} f(\sigma^2)$ so $\frac{s_n}{\sigma} \overset{P}{\to} 1$.