The paper deals with a linear integro-differential equation of order \(m\in N\) under linear boundary conditions \(Lu:=u^{(m)}+L^{(2)}u=f\) in (a,b), \(B_ eu=c_ e\), \(e=1,...,m\). This problem is approximated by a sequence of finite difference equations. The main result is to show the convergence \(\gamma_{r,q,h}\to \gamma_{r,q}\), which means that the conditions of the discretization \((L_ h,B_{eh}\) of \(L,B_ e)\) are, in the limit of sufficiently refined grids, equal to the conditions of the approximated problem.