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For any 0 < ϵ < 1, p ≥ 1 and each function $f\in L^p[0,1]$ one can find a function $g\in L^{\infty }[0,1)$ with mesx ∈ [0,1): g ≠ f < ϵ such that its greedy algorithm with respect to the Walsh system converges uniformly on [0,1) and the sequence $|c_k\left(g\right)|:k\in spec\left(g\right)$ is decreasing, where $c_k\left(g\right)$ is the sequence of Fourier coefficients of g with respect to the Walsh system.