If we have some $C_0$-semigroup $(S(t))_{t\geq 0}$ and $\lVert u(t)\rVert\leq C$ for $t\in [0,T)$ and $u(0)=u_0$, why do we then have, for $0<t<t'<T$, the estimation $$ \lVert (S(t)-S(t'))u_0\rVert\leqslant C\lvert t-t'\rvert? $$
This should be an easy consequence of the fact that $(S(t))$ is a $C_0$-semigroup but I do not see the reason.