The additive group of Zp is torsion-free

Write $m=p^{a}u$ with $a\ge 0$ and $p\nmid u$. Multiplication by $u$ is an automorphism of each cyclic group $\mathbb{Z}\mathrm{/}{p}^n\mathbb{Z}$, so from $mx=0$ it follows that $p^{a}x=0$. Assume for contradiction that $x\mathrm{\ne }0$, and choose the least index $t\ge 1$ with $x_t\mathrm{\ne }0$. By [F1], the earlier coordinates vanish and the tuple is compatible, so $x_t$ is represented by an integer divisible by $p^{t-1}$; because $x_t\mathrm{\ne }0$ in $\mathbb{Z}\mathrm{/}{p}^t\mathbb{Z}$, that representative is not divisible by $p^t$. Thus $x_t$ has exact $p$-adic divisibility $p^{t-1}$.

Compatibility propagates that exact divisibility to the coordinate $x_{t+a}\in \mathbb{Z}\mathrm{/}{p}^{t+a}\mathbb{Z}$, because reducing $x_{t+a}$ modulo $p^t$ gives the nonzero class $x_t$. Hence $p^a{x}_{t+a}$ is divisible by $p^{t+a-1}$ but not by $p^{t+a}$, so it is nonzero in $\mathbb{Z}\mathrm{/}{p}^{t+a}\mathbb{Z}$. This contradicts $p^{a}x=0$. Therefore $x=0$, and ${\mathbb{Z}}_p$ is torsion-free.