>>13780938>This is also an abrupt departure from the standard format and syllabus of analysis. The traditional course begins with a discussion of properties of the real numbers, moves
on to continuity, then differentiability, integrability, sequences, and finally infinite series,
culminating in a rigorous proof of the properties of Taylor series and perhaps even
Fourier series. This is the right way to build analysis, but it is not the right way to teach
it. It supplies little motivation for the early definitions and theorems. Careful definitions
mean nothing until the drawbacks of the geometric and intuitive understandings of
continuity, limits, and series are fully exposed. For this reason, the first part of this
book follows the historical progression and nloves backwards. It starts with infinite
series, illustrating the great successes that led the early pioneers onward as well as the
obstacles that stymied even such luminaries as Euler and Lagrange.
There is an intentional emphasis on the mistakes that have been made. These
highlight difficult conceptual points. 'fhat Cauchy had so much trouble proving the
mean value theorem or coming to terms \vith the notion of uniform convergence should
alert us to the fact that these ideas are not easily assimilated. The student needs
time with them. The highly refined proofs that we know today leave the mistaken
impression that the road of discovery in mathematics is straight and sure. It is not.
Experimentation and misunderstanding have been essential components in the growth
of mathematics.
A Radical Approach to Real Analysis
https://b-ok.cc/book/1182652/6098ac 
