Rate of convergence of a sequence in $\mathbb{R}^n$

Question

Let $d_k$ a convergent sequence in $\mathbb{R}^n$, with $d_k\to0$. Therefore, $$\forall \varepsilon>0 \ \ \exists \ \bar k : \forall k\ge\bar k, \ \|d_k\|<\varepsilon.$$ Suppose that if $\|d_k\|<\varepsilon$ then it holds: $$\|d_{k+1}\|\le\frac12\varepsilon+o(\varepsilon).$$

Then is it true that it exists an index $k_0$ from which $$\|d_{k+1}\|\le\frac12\|d_k\|?$$