A course dealing with the fundamental theorems of Infinitesimal calculus in a rigorous manner is now recognized as an essential part of the training of a mathematician.
The definition of a limit by means of the notion "value approached" has simplified the proofs of theorems, such as those giving necessary and sufficient conditions for the existence of limits....
It is hoped that the systematic use of the Heine-Borel theorem has helped materially toward this end since by means of this theorem it is possible to avoid almost entirely the sequential division or "pinching" process so common in discussions of this kind.
The general aim has been to obtain rigor of logic with a minimum of elaborate machinery.
The book may also be used as a basis for a rather short theoretical course on Real functions, such as is now given from time to time in some of our universities.
This little volume is designed as a convenient reference book for such courses; the examples which may be considered necessary being supplied from other sources.
It appears in the curriculum of nearly every university and is taken by students as "Advanced Calculus" in their last collegiate year, or as part of "Theory of Functions" in the first year of graduate work.
A course dealing with the fundamental theorems of Infinitesimal calculus in a rigorous manner is now recognized as an essential part of the training of a mathematician
