I am trying to do exercise 1.3 from Le Gall's Measure theory, probability and stochastic processes: Let $C([0,1],\mathbb{R}^d)$ be the space of all continuous functions from $[0,1]$ into $\mathbb{R}^d$, which is equipped with the topology induced by the sup norm. Let $\mathcal{C}_1$ be the Borel $\sigma$-field on $C([0,1],\mathbb{R}^d)$ and let $\mathcal{C}_2$ be the smallest $\sigma$-field on $C([0,1],\mathbb{R}^d)$ such that all functions $f\mapsto f(t)$, for $t\in [0,1]$, are measureable from $(C([0,1],\mathbb{R}^d), \mathcal{C}_2)$ into $(\mathbb{R}^d, \mathcal{B}(\mathbb{R}^d))$, where $\mathcal{B}(\mathbb{R}^d))$ is the Borel $\sigma$-field on $\mathbb{R}^d$. Show that $\mathcal{C}_2 \subset \mathcal{C}_1$.
Here is what I have tried: for a fixed $t\in [0,1]$ and $B\in \mathcal{B}(\mathbb{R}^d))$, let $X_t:C([0,1],\mathbb{R}^d) \rightarrow \mathbb{R}^d, f\mapsto f(t)$. Then by definiton $\mathcal{C}_2$ is generalted by the sets of the form $X_t^{-1}(B), t\in [0,1], B\in \mathcal{B}(\mathbb{R}^d))$. Hence to show $\mathcal{C}_2 \subset \mathcal{C}_1$, it suffcies to show that $X_t^{-1}(B)$ is open in $C([0,1],\mathbb{R}^d)$.
To this end, let $f\in X_t^{-1}(B)$, then $f(t)\in B$. To prove $X_t^{-1}(B)$ is open, it suffices to show that for each $f\in X_t^{-1}(B)$, there exists an open ball $B_{\epsilon}(f)=\{g\in C([0,1],\mathbb{R}^d)| \lVert g-f\rVert_{\infty} < \epsilon\}$ contained in $X_t^{-1}(B)$. Now since $B\in \mathcal{B}(\mathbb{R}^d)$, it is a union and countable intersection of open sets. Let $U_{f(t)}$ be the connected component of $B$ containing $f(t)$, then let $\epsilon$ be such that any function $g$ with $\lVert g-f\rVert_{\infty} < \epsilon$ satisfies $g(t)\in U_{f(t)} \subseteq B$, and the result follows.
I want to ask is there any issue with the above argument and whether there is any other proof. Thank you so much in advance.