pr.probability - Stein's method proof of the Berry-Esséen theorem

The relevant paper is "An estimate of the remainder in a combinatorial central limit theorem" by Erwin Bolthausen. I would like to understand the estimate on page three right before the sentence "where we used independence of $S_{n-1}$ and $X_n$":

$$begin{align}E|f'(S_n) - f'(S_{n-1})| &le E bigg(frac{|X_n|}{sqrt{n}} big(1 + 2|S_{n-1}| + frac{1}{lambda} int_0^1 1_{[z,z+lambda]} (S_{n-1} + t frac{X_n}{ sqrt{n}}) dtbig)bigg) \ &le frac{C}{sqrt{n}} big(1 + delta(gamma, n-1) / lambdabig)end{align}$$

that is, where $delta(gamma, n-1)/lambda$ shows up, which is the error term in the Berry-Esséen bound.

Here $S_n = sum_{i=1}^n X_i / sqrt{n}$ and $X1, ldots, X_n$ are iid with $E X_i =0$, $E X_i^2 = 1$, and $E|X_i|^3 = gamma$. Furthermore, denote $mathcal{L}_n$ to be the set of all sequences of $n$ random variables satisfying the above assumptions, then

$delta(lambda, gamma,n) = sup { |E(h_{z,lambda} (S_n)) - Phi(h_{z,lambda})|: z in mathbb{R}, X_1, ldots, X_n in mathcal{L}_n }$

and $h_{z, lambda}(x) = ((1 + (z-x)/lambda) wedge 1) vee 0$ and $delta(gamma, n)$ is a short hand for $delta(0,gamma, n)$, and $h_{z,0}$ is interpreted as $1_{(-infty, z]}$. I am mainly interested in verifying the second inequality, so I don't need to reproduce the definition of $f$ here, but it is related to $h$.

This paper is freely available online through springer. thanks in advance.

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