(I'm not sure whether this thread belongs in Pre- or the Uni forum)
How can we prove that for a discrete probability distribution
using the axioms of probability.
We know that since
but then what?
It is impossible to answer that question without knowing the complete set of axioms you have been given.
In many textbooks, Introduction to Probability by Laurie Snell, that is in fact part of the definition of probability densities on finite sample spaces.