Let $T = (T_t)_{t>0} $ be a strongly continuous semigroup (of bounded operators) on a Banach space $E$, i.e, $ \lim_{t \rightarrow z} \|T_tx- T_{z}x \|, \forall z >0, \forall x \in E $. Note that we do not assume strong continuity of $T$ at 0 and neither do we define $T(0)$. Let $$ w_0:= \inf_{t>0} \frac{\ln\|T_t \|}{t} = \lim_{t\rightarrow \infty} \frac{\ln\|T_t \|}{t} < \infty.$$
We call $w_0 \in [-\infty, +\infty)$ the type of $T$. Then for $w > w_0$ there exists $M_w$ such that
$$ \| T_t\| \leq M_w e^{wt}, \quad \forall t> 0 \quad (*)$$
This is Theorem 1.5.9 (page 18) in the book "Theory of semigroups and applications"- Sinha & Srivastava. The authors claim that we can find $t_0$ big enough such that
$$ \| T_t \| \leq e^{wt}, \quad \forall t > t_0. \quad (1.19) $$
This part, I can understand, then they said the fact that $t \mapsto \| T_t\|$ is bounded on any compact interval in $(0,\infty)$ (Lemma 1.5.3, page 14) together with (1.19) imply the existence of the number $M_w$ in $(*)$, but I don't see how it's done.
Any help is highly appreciated!