I'm trying to understand part of the proof of the Central Limit Theorem in Casella Berger 2E. I realize there are many functionally equivalent ways to prove the CLT, but there's a specific step the authors use that I'm trying to find justification for.
To summarize the salient parts, suppose we have a random variable $Y$ with mean 0 and variance 1, and its MGF exists for |$t$| < some positive $h$. We're interested in the limiting behavior of $M_Y(\frac{t}{\sqrt{n}})$, where $M_Y(t)$ is the MGF of $Y$. The Maclaurin expansion follows as:
$M_Y(\frac{t}{\sqrt{n}}) = 1 + \frac{(t/\sqrt{n})^2}{2} + R_Y(\frac{t}{\sqrt{n}})$
The authors then use $lim_{n\to\infty}\frac{R_Y(t/\sqrt{n})}{(t/\sqrt{n})^2}=0$ in the proof, and they appear to justify that step by citing later "Taylor's major theorem, which we will not prove here, is that the remainder from the approximation $g(x)-T_r(x)$ always tends to 0 faster than the highest-order explicit term".
Which result or formulation of Taylor's Theorem justifies this limiting behavior?