1. The first proof of the CLT is the "common" one (but probably in much-more simplified form) and should especially be read by AP- Stat teachers who have not seen a mathematical proof of the CLT at all in the past. 2. The second proof of the CLT is much-less known. However, those requesting it, which is relatively short and based on using L'Hospital's Rule to reach the conclusion, should read the first proof first as there were some things needed in it not repeated in the second proof. (It has only one "Preliminary" whereas the first has four.) 3. For those sans any math-stat background, you should not request all the proofs at once. Thus, request the first (as some did) and simply get back to me for more, if desired. 4. The two proofs that the chi-square stat is approximately chi- square distributed are much more "advanced" and so request such if you are comfortable with calculus beyond first-year calc. (Anyone teaching BC would likely be comfortable with the them...) 5. The proof the T-Stat does follow a t distribution (exactly if underlying population is normal) will be offered in the offing as the other results should be enough for now to read/study. (The theorem itself takes only a handful of lines to prove But is based on nine preliminary "Steps", which are primarily theorems. Thus, what I did was simply "put it all together" in a logical order whereas in a math-stat book such cannot be done.)
In general, these were written with the typical math teacher in mind, namely one who majored in math but didn't study mathematical statistics at all. As such, the two proofs of the CLT begin with "Preliminaries" such teachers may not be familiar with --- primarily defining a moment-generating function and the Uniqueness Theorem. Wrt the two chi-square test-stat proofs, calculus beyond the first two courses would probably be the appropriate prerequisite.
Thus, please make your requests accordingly --- if you want to read/ study more, then just get back to me with a "further request"...
-- David Bee
PS: If anyone wants to give a reading to the forthcoming proof I titled "Ten Steps Proving The Usual T Statistic Has a Student's t Distribution With n-1 Degrees of Freedom When Sampling From a Normal Population" as sort of a check or to make suggestions before such is posted, then please let me know, off-List of course, and I'll forward a copy to you. (The Theorem/Proof itself is of course Step 10.)