Simple Semigroup (of operator) inequality

Question

I have been able to show that the norm $\| x \|_{1} : = \sup_{t \geq 0} \| T(t)x \|$ is equivalent to the norm $\| x \|$, where $T$ is a bounded $C_0$-semigroup. I now want to show that $T$ is a contractive $C_0$-semigroup. That is, $\| T(t) \|_1 \leq 1$. I keep getting the inequality in reverse however, that is, $\| T(t) \|_1 \geq 1$:

$$\| x \|_{1} = \sup_{t \geq 0} \| T(t) x \| \leq \sup_{t \geq 0} \| T(t) \| \cdot \| x \| \leq \sup_{t \geq 0} \| T(t) \| \sup_{t \geq 0} \| T(t)x\|.$$ Can someone point out what I'm missing?

Answer 1

$$\|T(t)x\|_1 = \sup_{s \geq 0} \| T(s)T(t)x \| = \sup_{s \geq 0} \| T(s+t)x \| \leq \sup_{t+s \geq 0} \|T(t+s)x\| = \|x\|_1 $$