On Ruppert's preodering

For group topologies $\tau_1$, $\tau_2$ on a group $G$, $\tau_1 \le_c \tau_2$ means that every Cauchy ultrafilter on $(G, \tau_2)$ is a Cauchy ultrafilter on $(G, \tau_1)$, and $\tau_1 \le_b \tau_2$ means that for every neighborhood $V$ of the identity in $(G, \tau_1)$ there exists a~finite subset $K$ such that $KV$ is a~neighborhood of the identity in $(G, \tau_2)$. It is proved that the preorders $\le_c$ and $\le_b$ coincide.