For sequentially observed functional data exhibiting multiple change points in the mean function, we establish consistency results for the estimated number and locations of the change points based on the norm of the functional CUSUM process and standard binary segmentation. In addition to extending similar results in Venkatraman (1992) and Fryzlewicz (2014) for scalar data to the general Hilbert space setting, our main results are established without assuming the Gaussianity of the data, and under general linear process conditions on the model errors.