One intermediate form of convergence is called the Cauchy convergence. Here, we require the sequences of two different approximate solutions to approach arbitrarily close to each other,
While the pointwise convergence expression would imply the previous equation, it is important to note that the Cauchy convergence does not imply pointwise convergence as the functions could converge an answer other than the true solution.
While we cannot be assured pointwise convergence of these functions for all but the simplist cases, there do exist theorems that ensure that a sequence of approximate solutions must converge to the exact solution (assuming no computational errors) if the basis functions satisfy certain conditions. The theorems can only ensure convergence in an average sense over the entire domain, but it is usually the case that if the solution converges in an averge sense (energy, etc.), then it will converge in the pointwise sense as well.