Uniform convergence in the individual ergodic theorem for symmetric sequence spaces

It is proved that for any Dunford-Schwartz operator $T$ acting in the space $l_{\mathrm{\infty }}$ and for each $x\in c_0$ there exists an element $\widehat{x}\in c_0$ such that $\mathrm{\parallel }\frac{1}{n}{\sum }_{k=0}^{n-1}{T}^k(x)-\widehat{x}{\mathrm{\parallel }}_{\mathrm{\infty }}\to 0$.