Let's try to quantitfy our error a bit further. When we consider the preceding example, it seems to make sense that if we increase the number of degrees of freedom we used to approximate our function, the accuracy must approach the true solution. That is, we would hope that the sequence of approximate solutions will converge to the exact solution as the number of degrees of freedom (DOF) increases indefinitely.
This is a statement of pointwise convergence. It describes the approximate solution as approaching arbitrarily close to the exact solution at each point in the domain as the number of DOF increases.