Motivating the terminology of weaker vs stronger topology

Let $X$ be a set, $\tau_1,\tau_2$ two topologies on $X$, and consider the following statements

1. $\tau_1\subseteq \tau_2$ (i.e $\tau_1$ is coarser/weaker than $\tau_2$, or that $\tau_2$ is finer/stronger than $\tau_1$)
2. For every net $\langle x_i\rangle_{i\in I}$ in $X$ and $x\in X$, if $x_i\to x$ relative to the topology $\tau_2$ then $x_i\to x$ relative to the topology $\tau_1$.

It is clear to me that $(1)$ implies $(2)$. My question is whether the converse is also true, because then this would seem like a nice justification for the terminology "weak/strong" topology in the context of topological vector spaces.