As mentioned in this textbook by Morters and Peres on page 147:
Note that $D(a, b, t)$ is almost surely finite by the uniform continuity of Brownian motion on the compact interval $[0, t]$.
where $D(a, b, t)$ denotes the number of downcrossings of the interval $[a, b]$ before time $t$.
How do I prove that this is true?
(Note: The above quote is edited since this answer mentions that the "absolute continuity" originally in the book is a typo and should be replaced with "uniform continuity")