Proving calculus concepts with set theory/sequences

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.

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