Question about paper on Selmer groups

Question

Let $\textrm{Sel}_{n}E$ denote the $n$-Selmer group and $\textrm{Sel}_{p^{\infty}}E = \varinjlim_{n}\textrm{Sel}_{p^{n}}E$.

Proposition 5.10 of this paper http://arxiv.org/abs/1304.3971 states that

Suppose that $E$ is an elliptic curve over a global field $k$ with $E(k)_{\textrm{tors}} = 0$. Let $m$ and $n$ be positive integers such that $\textrm{char } k \nmid m, n$ and $m \mid n$. Then

1. The inclusion $E[m] \rightarrow E[n]$ induces an isomorphism $H^{1}(k, E[m]) \rightarrow H^{1}(k, E[n])[m]$.
2. This isomorphism identifies $\textrm{Sel}_{m}E$ with $(\textrm{Sel}_{n}E)[m]$.
3. If $p$ is a prime number and $e \in \mathbb{Z}_{\geq 0}$, then $\textrm{Sel}_{p^{e}}E \simeq (\textrm{Sel}_{p^{\infty}}E)[p^{e}]$.

My question is, where is the assumption $E(k)_{\textrm{tors}} = 0$ used in the proof? It seems the proof works with $E(k)_{\textrm{tors}}$ being anything.