How can Zeno's dichotomy paradox be disproved using mathematics? A brief description of the paradox taken from Wikipedia:

Suppose Sam wants to catch a stationary bus. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.
This description requires one to complete an infinite number of tasks, which Zeno maintains is an impossibility. This sequence also presents a second problem in that it contains no first distance to run, for any possible (finite) first distance could be divided in half, and hence would not be first after all. Hence, the trip cannot even begin.
The paradoxical conclusion then would be that travel over any finite
  distance can neither be completed nor begun, and so all motion must be
  an illusion.

How can this be disproved using math, as obviously we can all move a walk from one place to another?

We know that if Sam runs fast enough and long enough, he will eventually catch up to the bus. If both are moving at a constant speed, there is no need to decompose their motion into infinitely many, ever decreasing intervals. A simple application of the speed-distance-time formula will tell us that Sam will catch up to the bus in $\frac d{s_2 - s_1}$ seconds where $d$ = the head start by the bus (m), $s_1$ = the speed of the bus (m/s), and $s_2$ = Sam's speed (m/s). 
In any finite time interval, we know that Sam and the bus with pass through infinitely many points in space, with an event being associated with their arrival at each point. To the modern mind, there is nothing "paradoxical" or even counter-intuitive about this.
Historical note: It wasn't until Galileo's pioneering efforts in physics and the introduction of the scientific method several centuries after Zeno and Aristotle that we were able to actually measure the speed of an object.
