If $x_0,h$ belong to a metric vector space and $ε>0$, is there always a sequence $(t_n)_{n∈\mathbb N}\subseteq\mathbb R$ with $x_0+t_nh\in B_ε(x_0)$?

Question

Let $(X,d)$ be a metric $\mathbb R$-vector space, $x_0,h\in X$ and $\varepsilon>0$. If $d$ is induced by a norm $\left\|\;\cdot\;\right\|$, then $$x_0+th\in B_\varepsilon(x_0)\;\;\;\text{for all }|t|<\frac\varepsilon{\left\|h\right\|}.$$ However, if $d$ is not induced by a norm, can we still show that there is a $(t_n)_{n\in\mathbb N}\subseteq\mathbb R\setminus\{0\}$ with $$x_0+t_nh\in B_\varepsilon(x_0)\;\;\;\text{for all }n\in\mathbb N\tag1$$ and $t_n\xrightarrow{n\to\infty}0$?