Solution
= Solution
The domination hypothesis includes $r=0$. For any approximation error $h$, <linearity> and the <triangle inequality> therefore give the uniform estimate
$$
|T_rh-T_0h|\leq|T_rh|+|T_0h|\leq2S(h).
$$
For a general input $g$ and an approximating $f$, this implies
$$
|T_rg-T_0g|
\leq|T_rf-T_0f|+2S(g-f).
$$
The first term converges on the dense class, while the <weak type (1,1)> estimate controls the exceptional set where the second is large.