I was reading Silverman's Arithmetic of Elliptic Curves I have a question on computing the Mordell-Weil group of an elliptic curve over $E(\mathbb{Q})$.
Adapting the argument given in Silverman we have that we successive compute $$ S^{(2)}(E/\mathbb{Q}) = S^{(2, 1)}(E/\mathbb{Q}) \supset S^{(2, 2)}(E/\mathbb{Q}) \supset S^{(2, 3)}(E/\mathbb{Q}) \supset \ldots $$ and $$ T_{(2,1)}(E/\mathbb{Q}) \subset T_{(2,2)}(E/\mathbb{Q}) \subset T_{(2,3)}(E/\mathbb{Q}) \subset \ldots $$ where $T_{(2,r)}(E/\mathbb{Q})$ generated by the set $$ W = \{ P \in E(\mathbb{Q}) \; | \; h(P) \leq r \}. $$
Why is it that once we have that $$ S^{(2, m)} (E/\mathbb{Q}) = T_{(2, r)} (E/\mathbb{Q}), $$ then $2^{m-1}Ш(E/\mathbb{Q})[2^m] = 0$?