Let $X$ be a Banach space and $\{A_t\}_{t\in R}$ a family of linear and continuous maps $X \rightarrow X$ such that function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous for any $x \in X$. Show that $\sup\{\|A_t\|:t\in K\}$ is finite for any compact subset $K \subset \mathbb{R}$.
My attempt
Function $\mathbb{R} \ni t\rightarrow \|A_t x\| \in \mathbb{R}$ is continuous so it is bounded on any compact $K \subset \mathbb{R}$.
From Banach-Steinhaus it follows that $\sup_{t\in K}\|A_t\|<\infty.$
Indeed, it follows from Banach-Steinhaus applied to the family $(A_t)_{t\in K}$ for a fixed compact $K$.
We have to point out that $\sup_{t\in K}\lVert A_t(x)\rVert$ is bounded for any $x\in X$.