A Remark on Zeros of Brownian Motion

Let $\{W(t), t\ge 0\}$ be a standard Brownian motion. If $I$ is a bounded interval on which $W$ has no zero, an almost sure lower bound to $\inf\{|W(t)|, t\in I\}$ can be provided, when $I$ is taken from a given countable family of intervals covering the positive half-line.