I am struggling with a detail related to the trajectories of stochastic processes. I am working with Brownian motion, but technically any continuous stochastic process would make no difference. However, to have a specific model in mind, let's say that it is a Brownian motion starting at $0 $ and the let $P^0 $ be the probability measure associated with a Brownian motion that starts at $0. $
I am working with the canonical stochastic process so that the space is $\Omega \doteq C[0, \infty) $ and each $\omega \in \Omega $ is a continuous function on $[0, \infty). $ In a paper I found a statement that says that: $$ P^0[\{\omega \in \Omega: \omega(1/n) = 0\}] = 0. $$
If this fact is true, then one can prove that the trajectories or Brownian motion that are constant on the interval $[0, \epsilon] $ for some $\epsilon > 0 $ have probability zero as well.
I cannot convince myself that constraining a single coordinate of the process causes it to have probability zero. Could anyone help me understand why it is so? I believe it is as I saw this fact mentioned directly or indirectly in more than one place.
Thank you