RU-ALLRUSSIAN 1986 Problem 10

Show that for all positive reals $\mathbf{x}_{\mathrm{i}}$ we have $1 / \mathbf{x}_{1} + 1 / (\mathbf{x}_{1} + \mathbf{x}_{2}) + \ldots +\mathbf{n} / (\mathbf{x}_{1} + \ldots +\mathbf{x}_{\mathrm{n}}) < 4 / \mathbf{a}_{1} + 4 / \mathbf{a}_{2} + \ldots +4 / \mathbf{a}_{\mathrm{n}}$ .