Close points on a Lie Group with Left-Invariant Metric

Question

Let $G$ be a Lie group with left-invariant metric $d$ coming from a left-invariant Riemannian metric. If $a \in G$ is close to the identity $d(a,e) < \epsilon$, is $ab$ close to $b$ in general? For $b$ fixed, $a_n \to a$ implies that $a_nb \to ab$ by continuity. But is there something stonger like $d(ab,b) < C_b\epsilon$ for some constant $C_b$ that maybe depends on $b$ in some explicit way?