It has been known that the conditional Lindeberg condition and the fact that the conditional variance of a martingale is equal to 1 imply the weak convergence of the martingale to the standard normal law. It was not known whether the conditional Lindeberg condition is necessary or not until \textit{A. G. Sholomitskij} [Theory Probab. Appl. 43, No. 3, 434-448 (1998); translation from Teor. Veroyatn. Primen. 43, No. 3, 490-508 (1998; Zbl 0954.60016)] asserted it holds true by adding additional constraints on the martingales. The proof in Sholomitskij's paper seems to be complicated. In this article the author presents a simple and short proof of the necessity of the conditional Lindeberg condition for normal convergence of martingales.