On the uniqueness of a Walsh series converging on subsequences of partial sum

We show that if a Walsh series whose coefficients tend towards zero is such that the subsequence of its partial sums indexed by $n_k$, where $n_k$ satisfies the condition $2^{k-1}<n_k\le2^k\quad(k=0,1,2,\dots)$, tends everywhere, except possibly for a denumerable set, towards a bounded function $f(x)$, then this series is the Fourier series of the function $f(x)$.