Proving calculus concepts with set theory/sequences

Cinitiator

1. The problem statement, all variables and given/known data
How is the field of mathematics which focuses on defining the calculus concepts (such as limits) using the set theory and/or sequences called? Where numbers are seen as discrete, and concepts like limits are defined in these terms.


2. Relevant equations



3. The attempt at a solution
Googling, without much success.

Undoubtedly0

Generally I believe this field is known as mathematical analysis, or simply analysis (see http://en.wikipedia.org/wiki/Mathematical_analysis).