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?

It can't. It's not a mathematical statement, it's a statement about the nature of physical space.

At least for the first problem, the obvious mathematical answer is that the "total distance" is finite, because it's the infinite sum $\sum 2^{-n}$, which converges. But the whole point of the paradox is that it's making a statement about the physical world. It's philosophically difficult to say whether or not the above infinite series argument can really be applied to physical space. In particular, is it even meaningful to subdivide a physical length indefinitely? Are physical lines fundamentally continuous or discrete? Do any of these questions really mean anything?

No matter how far you postpone it, at some point you're going to have to cross the bridge from the mathematical model into the real world, and that will always be a philosophical problem, not a mathematical one.

The implicit assumption here is that 1. cutting distances into infinitely many pieces is different than cutting times into infinitely many pieces, and/or 2. an infinite sum cannot converge. Neither of which are true.

The sum of distances $1/2+1/4+1/8+1/16...$ equals $1$ as expected. We must also split time up into correspondingly small steps: adding intervals of $1/2+1/4+1/8...=1$ (possibly scaled for the appropriate speed), which also sums to $1$. The times add to a finite time in the exact same way as the distances add to a finite distance. The claim that one cannot complete infinitely many tasks implicitly assumes that infinitely many smaller and smaller tasks cannot add together into one well-defined task that takes finite time, which is not true.

One could of course instead reject the idea that distance and time can be split infinitely in this way at all, claiming that actual motion cannot be split in this way and that the difference between this thought experiment and reality rests crucially on that.

Zeno may want us to infer that the time necessary to complete these infinite number of tasks is infinite. However, he omits any mention of the speed at which the traveler is moving. There's nothing in this paradox that says the traveler can't move at a constant speed, which simply means that the time taken to move a given distance is proportional to the distance.

Whether Zeno understood infinite sums and convergence would be interesting background to how he arrived at his conclusion, but it's irrelevant to the mathematics known today.

So what's obvious mathematically is that the infinite sum of the distances from these infinite number of tasks is still a finite distance and (for a traveler moving a constant speed) the time it takes to travel that distance is proportional and therefore finite.

The same conclusion can be reached even if the speed is not constant, and may be answered using calculus, which Zeno wasn't familiar with.

To travel any distance, a traveler must not take the path that Zeno took. There are several responses to your question that begin with Zeno's original perceptions as if they are somehow entrenched canon in philosophy (in understanding physical nature) and that one must start there to begin to answer the OP's question. But to start there is just as fruitless as traversing the distance in an infinite number of individual tasks, where even the first task (of allowing the traveller to traverse that first infinitesimal distance) is hobbled by awkward concepts on motion.

This one's easy; sequences don't have to have a "first" element, nor does any particular term in a sequence have to have a "next" element.

This "paradox" is not really any different from being confused about the fact that the integers do not have a smallest element, nor the fact that in the extended integers, the element $-\infty$ does not have a successor; the confusion is just disguised better.

We often label points in a sequence with natural numbers, as this is the most common use case for the notion of a sequence, and thus are in the habit of thinking any sequence must have a first element, and every other point has a predecessor, and conversely every point is either last or it has a successor.

However, if we work with sequences that cannot be labeled in such a way -- e.g. marking the midpoint, the quarter point, the one-eighth point and so forth of our journey, along with marking the two endpoints, and observe that we have to transverse them in order -- we can make grave errors if we treat them as if they can be.