Geometric bound for omitted Haar details (${\sum }_{j=J}^K\delta /\sqrt{L{2}^{-j}}\le \frac{\delta 2^{K/2}/\sqrt{L}}{1-2^{-1/2}}$)

At each point there is at most one supported Haar wavelet per scale. If a detail coefficient is below $\delta$, its value there is at most $\delta 2^{j/2}/\sqrt{L}$. Summing these bounds up to level $K$ gives the displayed geometric estimate. Selecting $K$ so the final term is of the order of the modulus of continuity controls all omitted fine details, even though arbitrary deletions from a uniformly convergent series need not preserve convergence.