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?

Even if we allow for successively smaller intervals of time, and ignore the infinite sum having a finite answer, we can introduce some physics and bring it into the real world.
I'll introduce a concept called Planck's Length : 1.61619926 × 10^-35 meters.
This is the smallest measure that exists in reality.  It is the smallest distance you can travel.  If you like, it is the Pixel Size of reality.  
The Corollary is Planck Time : 5.39106(32) × 10−44 s
Which is the smallest time in which anything can happen (It's the time it takes light to travel one Planck length). You could consider it the clock rate of reality.
So while mathematics is happy to allow our two running sums (time and distance) to get successively closer but never touching the end, Physics dictates that eventually, you can't divide by two, because distance and or time can't be measured that finely, and eventually, you run into the bus.
The impact of this is several paradoxical concepts, such as Zeno's (from the OP,) or  Gabriel's horn (finite volume, infinite surface area) suddenly collapse in the face of reality.  Or, if you like, dx and dt cease to exist as infinitesimals, and just become the smallest possible Delta-x and Delta-t.
An interesting philosophical upshot of Planck Length and Planck Time is that reality as we know it, could actually be a simulation running on a real computer somewhere.
