Suppose that $(T(t))_{t\geq0}$ is a $C_0$-semigroup on a Banach space $X$ and assume that there exists $x\in X$ such that $\{T(t)x:\ t\geq0\}$ is dense in $X$. I wonder why the set $\{T(t)x:\ t\geq t_0\}$ is dense for every $t_0>0$. I do not see the reason why this should be true. I tried this with a sequence argument, but it doesn't work. Is there any idea? Maybe by contradiction? Thanks a lot.

Suppose that $(T(t))_{t\geq0}$ is a $C_0$-semigroup on a Banach space $X$ and assume that there exists $x\in X$ such that $\{T(t)x:\ t\geq0\}$ is dense in $X$. I wonder why the set $\{T(t)x:\ t\geq t_0\}$ is dense for every $t_0>0$. I do not see the reason why this should be true. I tried this with a sequence argument, but it doesn't work. Is there any idea? Maybe by contradiction?