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The undergraduate books in probability theory that I know do not provide a rigorous proof of even a weak version of the central limit theorem. Instead, they rely on Lévy's continuity theorem, whose proof they choose to omit due to it allegedly being too technical. It seems (see comments here) that the proof of the De Moivre–Laplace theorem which is just a special case of the central limit theorem is not as difficult to prove and I've been searching for a sufficiently rigorous proof. However, in order to prove it everyone either refers to the central limit theorem which they as aforementioned do not provide a proof of or they provide some non-rigorous proof.

I've searched on google for quite a while now and I am unable to find a proof that is rigorous and does not refer to the central limit theorem. This makes me think that a proper proof of the De Moivre–Laplace theorem is beyond the scope of undergraduate mathematics students as well.

Is this correct? If not, how does one prove this thing? Preferably I'd like it to be proven with some ordinary real analysis methods if possible. If I am to do it by myself, where do I start? And no, I don't like the wikipedia proof.

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    Volume one of Feller's book has a proof, but I can't say whether it will meet your standards for rigor. – 2017-01-10
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    @carmichael561 I will have a look at it. – 2017-01-10
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    @carmichael561 I can't follow the notation of that proof since it is assumed that the reader has gone through the previous content and gotten used to it. I assume http://ruangbacafmipa.staff.ub.ac.id/files/2012/02/An-Introduction-to-probability-Theory-by-William-Feller.pdf was the book you were refering to yes? – 2017-01-10
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    Yes, that's the book I was referring to. – 2017-01-10
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    This is exactly what Kai Lai Chung is doing in his book *Elementary probability theory with stochastic processes* (Springer, 1974, Undergraduate texts in mathematics), see Theorem 6 (stated on pages 215-216) in Section 7.3. – 2017-01-10
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    @Did I will look it up a bit later, thank you. – 2017-01-10
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    @Did I will keep this in mind for when I have the money. At the moment I will keep searching for something free. – 2017-01-10
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    You can save your money... – 2017-01-10

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Excerpts from Kai Lai Chung's book Elementary probability theory with stochastic processes (Springer, 1974, Undergraduate texts in mathematics):