### Limit Sum Theorem

In this article, we derive the classic limit sum theorem without using the Axiom of Choice. Elementary theorems from arithmetic, such as the Triangle Inequality, are referenced and introduced as needed. In certain cases, a number of easy, simplistic steps using theorems of logic are compressed into a single step. As will be shown, the key to the whole process is the tactical use of the logical theorems for importing quantifiers.

We begin with a single Deduction from Premises (DFP) and once that is finished, the remaining deduction is a sequence of theorems taking small steps that eventually get us to the limit sum theorem. Any sentence that is derivable from the axioms of logic and arithmetic is labeled "Th", that is, "Theorem". The vast majority of these sentences are intermediate results and would not be labeled as theorems in traditional writing. The value of a function such as F evaluated at x is simply written as "Fx", that is, brackets are omitted. "Pms" indicates a premise in a DFP.

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