Brownian paths are not uniformly continuous on the half-line
= Brownian paths are not uniformly continuous on the half-line
{c}
For every fixed mesh $1/k$, <independent increments> with the <normal distribution> imply increments of absolute value greater than one occur infinitely often <almost surely>, by the second <Borel-Cantelli lemma>. A countable intersection over $k$ contradicts <uniform continuity> on $[0,\infty)$.