I would like to show that if $X_t$ is a continuous and adapted stochastic process (real valued) then it is progressively measurable.
Here is my attempt : consider $B\in\mathcal{B}(\mathbb{R})$. Then we have
$$ X^{-1}(B) =\big\{ (\omega,t)\in\Omega\times [0,T] : X(\omega,t)\in B\big\} = \big\{ \omega\in\Omega : X(\omega,.)\in B\big\}\times[0,T]\cap\Omega\times \big\{ t\in[0,T] : X(.,t)\in B\big\}\in\mathcal{F}_t\otimes\mathcal{B}([0,T]) $$
Using the fact that $X$ is continuous in $t$ (at fixed $\omega$ and hence measurable from $([0,T],\mathcal{B}([0,T])$ to $(\mathbb{R}, \mathcal{B}(\mathbb{R}))$ the set $\big\{ t\in[0,T] : X(.,t)\in B\big\}$ is in $\mathcal{B}([0,T])$.
A similar argument when $t$ is fixed shows that the set $\big\{ \omega\in\Omega : X(\omega,.)\in B\big\}$ is in $\mathcal{F}_t$ since $X_t$ is adapted.
That was my idea. I would like to know if it is correct please and if not what can I improve ?