We consider the problem of non-negative super-resolution, which concerns reconstructing a non-negative signal x = Sigma(k )(i=1)a(i)delta(ti) from m samples of its convolution with a window function phi(s - t), of the form y(s(j)) = Sigma(k)(i=1) a(i) phi(s(j) - t(i)) + delta(j), where delta(j) indicates an inexactness in the sample value. We first show that x is the unique non-negative measure consistent with the samples, provided the samples are exact. Moreover, we characterise non-negative solutions (x) over cap consistent with the samples within the bound Sigma(m)(j=1) delta(2)(j) <= delta(2). We show that the integrals of (x) over cap and x over (t(i) - epsilon, t(i) + epsilon) converge to one another as epsilon and delta approach zero and that x and (x) over cap are similarly close in the generalised Wasserstein distance. Lastly, we make these results precise for phi(s - t) Gaussian. The main innovation is that non-negativity is sufficient to localise point sources and that regularisers such as total variation are not required in the non-negative setting. (C) 2019 Elsevier Inc. All rights reserved.