I'm looking at problem 4.2 in "Brownian Motion and Stochastic Calculus" by Karatzas and Shreve. The goal is to show that on $C[0,\infty)$, the Borel sigma algebra generated by "topology of local uniform convergence" is equivalent to the product sigma-algebra generated by the Cylinder sets.
The so-called "topology of local uniform convergence" is induced by the metric $$d(f,g):=\sum_{n}\frac{1}{2^{n}}\min\left\{ 1,\sup_{s\in\left[0,n\right]}\left\Vert f(s)-g(s)\right\Vert \right\}$$
I'm stuck on showing that the projections $\pi_{t}(\omega)=\omega(t),\; \omega\in C[0,\infty),$ are continuous. (Such that they're Borel-measurable.)
In the cases where $t\leq1$, we get: $$ \begin{align*} \left\Vert \pi_{t}(\omega)-\pi_{t}(\psi)\right\Vert &=\left\Vert \omega(t)-\psi(t)\right\Vert \\ &\leq\sup_{s\in\left[0,n\right]}\left\Vert \omega(s)-\psi(s)\right\Vert \\ &=\sum_{n}\frac{1}{2^{n}}\sup_{s\in\left[0,1\right]}\left\Vert \omega(s)-\psi(s)\right\Vert \end{align*} $$ So here we even have Lipschitz continuity. But I'm struggling to extend this.
If $t$ is any non-negative number, then consider $n$ such that $n\geqslant t$ and notice that $$\left\Vert \pi_{t}(\psi_1)-\pi_{t}(\psi_2)\right\Vert \leqslant \sup_{s\in\left[0,n\right]}\left\Vert \psi_1(s)-\psi_2(s)\right\Vert\leqslant 2^n\underbrace{2^{-n}\left\Vert \psi_1(s)-\psi_2(s)\right\Vert}_{\leqslant d(\psi_1,\psi_2)},$$ hence $$\left\Vert \pi_{t}(\psi_1)-\pi_{t}(\psi_2)\right\Vert \leqslant 2^nd(\psi_1,\psi_2)$$ from which the continuity of $\pi_t$ follows.