The Haar system in $L_1$ is monotonically boundedly complete

Answering a question posed by J. R. Holub we show that for the normalized Haar system $\{h_n\}$ in $L_1[0,1]$ whenever $\{a_n\}$ is a sequence of scalars with $|a_n|$ decreasing monotonically and with $\sup_N\|\sum_{n=1}^N a_n h_n\| < \infty$, then $ \sum_{n=1}^\infty a_n h_n$ converges in $L_1[0,1]$.