Let \[ B_ n(f; x)= \sum^ n_{k=0} f\left({k\over n}\right)\left(\begin{smallmatrix} n\\ k\end{smallmatrix}\right) x^ k(1-x)^{n- k} \] and \(w_ \varphi(f; \delta)= \sup_{0\leq t\leq \delta} \sup_ x| f(x- t\varphi(x))- 2f(x)+ f(x+ t\varphi(x)))|\), where \(f\in C[0,1]\), \(\varphi(x)= \sqrt{x(1-x)}\) and the second supremum is taken for those values of \(x\) for which every argument belongs to \([0,1]\). Concerning degree of approximation the following inequality is well-known [\textit{Z. Ditzian} and the author, Moduli of smoothness (1987; Zbl 0666.41001)]: \[ \| B_ n(t)- f\|\leq Cw_ \varphi\left(f; {1\over \sqrt n}\right). \] However it remained an open problem whether its inverse holds only very recently [\textit{Z. Ditzian} and \textit{K. G. Ivanov}, J. Anal. Math. 61, 61-111 (1993; Zbl 0798.41009)] showed that \[ w_ \varphi\left(f; {1\over \sqrt n}\right)\leq C{m\over n}(\| B_ n(f)- f\|+ \| B_ m(f)- f\|) \] holds for \(m\geq kn\), \(n= 1,2,\dots\)\ . In particular \[ w_ \varphi\left(f; {1\over \sqrt n}\right)\sim\| B_ n(f)- f\|+ \| B_{kn}- f\|, \] where \(\sim\) means that the ratio of the two sides lies in between two positive constants independently of \(f\in C[0,1]\) and \(n\). They also conjectured that the second term can be dropped. In this paper the author shows that their conjecture is true. In other words he proves that \[ \| B_ n(f)- f\|\sim w_ \varphi\left(f; {1\over \sqrt n}\right). \] He first proves a result for Szász-Mirakyan operators and then indicates how the above result can be established. The proof is long and involves delicate analysis.