A fixed finite modified Hamiltonian is close to the original energy

If $K=\mathrm{\varnothing }$, every $M_j=0$ and hence $C_0=0$. The pointwise estimate has no instances, and every nonnegative real is a uniform upper bound for the empty error family, so its least such bound is $0$. Assume henceforth $K\mathrm{\ne }\mathrm{\varnothing }$. By [F1], every $H_j$ is continuous; [F2] applied to $\mathrm{\mid }{H}_j\mathrm{\mid }$ gives its finite nonnegative maximum $M_j$ on $K$, equal to the corresponding supremum. Their finite sum $C_0$ is therefore finite and nonnegative.

For $z\in K$, [F1] and step 1.1 give $\mathrm{\mid }{H}_h^{[N]}(z)-H(z)\mathrm{\mid }\le {\sum }_{j=r}^N{h}^j{M}_j\le h^r{\sum }_{j=r}^N{M}_j=C_0{h}^r$ for $0<h\le 1$, since every $j\ge r$ has $h^j\le h^r$. The error function is continuous on nonempty compact $K$, so [F2] also gives its attained maximum and ordinary real supremum, both bounded by this same right-hand side. Its nonnegative maximum is its least nonnegative upper bound. Together with the empty branch in step 1.1, this proves every stated case without an empty real supremum.