Brownian Motion and Continuity Consider a Brownian Motion $(B_{t})_{t\geq0}$. In my lecure notes it says, without proof, that $\mathbb{P}\left(\sup_{t,s\leq N}\left\{ \left|B_{t}-B_{s}\right|:\left|t-s\right|<\delta\right\} <\varepsilon\right)$ converges to $0$ for $\delta$ tending to $0$.
I think it is a consequence of the continuity of the sample paths, but can anyone help with a proof?

First there's a typo: you should have $>\varepsilon$ or should say that the probability converges to to $1$, not $0$.
Anyway, here's my proof. It's informal and skips over details; you should fill these in.
Fix $t$ and $s$ for a moment. Then your probability is the probability that a normal variable centered at zero whose variance is at most $\delta$ is outside the interval $(-\varepsilon,\varepsilon)$. This goes to zero by Markov's inequality (or Chebyshev's inequality, different authors seem to use different names for this).
Now fix $s$ and let $t \geq s$ be arbitrary. Then the Markov property tells us that $B_t - B_s$ is another Brownian motion, call it $B'_{t-s}$. Doob's maximal inequality estimates your probability by a constant times the corresponding probability for $B'_{N-s}$, which goes to zero by the previous argument.
Now allow $s$ to vary as well. Then the probability still goes to zero, because the suprema over $t$ from the previous paragraph are bounded by our estimate for the supremum over $t$ with $s=0$.
