A bounded operator $T$ on a Banach space is said to be power bounded if $\sup_{n\geq 0} \|T^n\|<\infty$.
Show that a $C_0$-semigroup $(T(t))_{t\geq 0}$ is bounded if and only if there exists $t> 0$ such that $T(t_0)$ is a power bounded operator.
My attempt: One implication is obvious. Conversely, let $T(t_0)$ be power bounded. Then there exists $M>0$ such that $$M=\max \{\sup_{0\leq t\leq t_0} \|T_t\|, \sup_{n\geq 0} \|T(nt_0)\|\}.$$ Now, let $t\geq 0$. If $t\leq t_0$, then $\|T(t)\|\leq M$. If $t>t_0$, then there exists $n\in\mathbb N$ such that $t=nt_0+\tau$ for some $\tau<t_0$. Thus $$\|T(t)\|\leq \|T(nt_0)\|\ \|T(\tau)\|\leq M^2.$$
It follows that $\|T(t)\|\leq \max\{M,M^2\}$ for all $t\geq 0$ and is hence, bounded.
Is my proof correct?