The truncation error of a consitent scheme can be made abitrarily small by a suffiecient reduction of the increments \(\Delta x\;and\;\Delta t\).
Although, this cannot be expected for the error of the numerical solution: \(u^n_j-u(j\Delta x, n \Delta t)\)
Definition: If the error of the numerical solution approaches zero as the grid is refined, the solution is called convergent. If this holds for any initial conditions, than the scheme is called convergent too.
Consistency od a scheme does not guarantee convergence!
For convergence of a scheme, the initial conditions needs to lie within the domain of dependency.
For the advection equations (Euler forward and upstream) this is the case if the CFL criterion holds. That is:
\(c\Delta t \leq\Delta x \)
(If a scheme is convergent it is also stable!)
consistency plus stability gives convergence!