On an estimate of the remainder in Lindeberg's theorem

Let $X_1,X_2,\dots$ be a sequence of independent random variables which have the distribution functions $F_1(x),F_2(x),\dots$, the mean values $m_1,m_2,\dots$, the finite variances $\sigma_1^2,\sigma_2^2\dots$ and infinite absolute moments of order $2+\delta$ for any $\delta>0$. The examples of sequences are given for which the estimate
$$\sup_x|F_n(x)-\Phi(x)|\le C\Psi_n(\varepsilon s_n)$$
does not hold true. Here $C$ is a constant, $\varepsilon$ is any fixed positive number and $F_n(x)$, $\Phi(x)$, $\Psi_n(\varepsilon s_n)$ are defined on p. 141.