Let $X$ be any distribution with defined mean, $\mu$, and standard deviation, $\sigma$. The central limit theorem says that $$ \sqrt{n}\frac{\bar{X} - \mu}{\sigma} $$ converges in distribution to a standard normal distribution. If we replace $\sigma$ by the sample standard deviation $S$, is there a theorem stating that $$ \sqrt{n}\frac{\bar{X} - \mu}{S} $$ converges in distribution to a t-distribution? Since for large $n$ a t-distribution approaches a normal, the theorem, if it exists, may state that the limit is a standard normal distribution. Hence, it would seem to me that t-distributions are not very useful - that they are useful only when $X$ is roughly a normal. Is this the case?
If it is possible, would you indicate references that containing a proof of this CLT when $\sigma$ is replace by $S$? Such a reference could preferably use measure theory concepts. But anything would be great to me at this point.