Clarify my understanding for central limit theorem from a statement Asked what the central limit theorem says, a student replies, "as you take larger and larger samples from a population, the histogram of the sample values looks more and more Normal". Is the student right? Explain your answer.
My answer the student is wrong because the histogram of the sample values will look like the population distribution, whatever that distribution might look like as the sample size increases. The CLT says the sample mean follows a normal distribution with mean $u$ and variance $σ^2/n$ as the sample size goes to infinity. But CLT fails to population
that has fat tails such as Cauchy Distribtion. 
Is this right? If so, do you think I could add a little more? 
 A: The Central Limit Theorem examines sums of random variables from which we subtract the mean of the sum and then divide the whole by the standard deviation of the sum: let $Y_1,...,Y_n$ be random variables and define $S_n \equiv \sum_{i=1}^nY_i$. Then the CLT examines the random variable
$$Z_n = \frac {S_n - E[S_n]}{\sqrt {\text {Var}(S_n)}}$$
and what is its limiting distribution.
Variants of the CLT materialize as we increase the number of assumptions: the most basic variant is the assumption that the $Y_i$'s are identically and independently distributed, with existing / finite moments (but a CLT exists even for distributions that do not have finite variance, for a taste see here).
In most variants, the CLT provides sufficient but not necessary conditions for it to hold.
If these conditions are satisfied, the usual CLT asserts that $Z_n \rightarrow_d N(0,1)$.  
So in order to be able to apply the CLT, one must in the first place be examining a $Z_n$-variable. 
Note that the work around (if the CLT holds)
$$Z_n \rightarrow_d N(0,1) \Rightarrow S_n \sim_{approx.} N\left(E[S_n],\text {Var}(S_n)\right)$$
is only an approximation, and for "large" but finite $n$.  
The "student's answer" is wrong, not because it does not refer to "the sample mean", but because one cannot define a "population" in such a way so as drawing samples from it and examining their empirical distribution function becomes equivalent to examining the limiting distribution of a $Z_n$ variable as $n\rightarrow \infty$ (the latter being what the CLT does).
