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.