Starting from \(x=0\), a particle makes independent jumps of length

\[\Delta x=\sigma \sqrt{\Delta t}\]

to the right or to the left every \(\Delta t\) time units. The respective probabilities of jumps to the right and to the left are

\[p=\frac{1}{2}\left(1+\frac{\mu}{\sigma} \sqrt{\Delta t}\right) \text { and } 1-p \text { with } \sqrt{\Delta t} \leq\left|\frac{\sigma}{\mu}\right|, \sigma>0\]

Show that as \(\Delta t \rightarrow 0\) the position of the particle at time \(t\) is governed by a Brownian motion with drift with parameters \(\mu\) and \(\sigma\).