We study the convergence of a Cartesian method for elliptic problems with immersed interfaces. This method is based on additional unknowns located on the interface, used to express the jump conditions across the interface and discretize the elliptic operator in each subdomain separately. It is numerically second-order accurate in \(L^{\infty }\) -norm. We prove the convergence of the method in two cases: the original second-order method in one dimension, and a first-order version in two dimensions. The proof of convergence takes advantage of a discrete maximum principle to obtain estimates on the coefficients of the inverse matrix. More precisely, we obtain estimates for the sums of the coefficients of several blocks of the inverse matrix. Associated to the consistency error, which has different leading orders throughout the domain, these estimates lead to the convergence results. The methodology exposed in the article allows to take into account the effects of different orders of approximation errors across the domain and their effective influence on the total convergence order.