Explore the concept of real numbers and Cauchy sequences of rationals in this 36-minute mathematics lecture. Delve into the reasons why pure mathematicians cling to the idea of real numbers and examine whether they truly support 'real values' or are intrinsically approximate notions. Revisit a crucial calculation by Archimedes to motivate the discussion. Investigate the definitions of length, area, and function values, and analyze the common aspects of various types of measurements. Challenge your understanding of fundamental mathematical concepts and gain insights into the nature of real numbers and their applications in pure mathematics.