The discussion centers around the justification for canceling out the dx in the context of u-substitution during integration. Participants explore the relationship between differentials, derivatives, and the notation used in integrals, particularly in relation to the Riemann sum concept.
Participants express a mix of agreement and disagreement, particularly regarding the mathematical validity of infinitesimals and the intuitive understanding of calculus concepts. There is no consensus on the acceptance of infinitesimals in the mathematical community.
Limitations in understanding arise from the intuitive notion of infinitesimals versus formal definitions involving limits. The discussion reflects varying levels of comfort with the terminology and concepts related to calculus.
Hmm, color me skeptical. Why can't you explicitly construct a sequence satisfying x_n \in (x - \frac{1}{n}, x + \frac{1}{n})? For instance, x_n can be the point a third of the way into the interval (x - \frac{1}{n}, x + \frac{1}{n}).micromass said:This is where choice is used. You can never give exact definition of the x_n. This is why it requires choice.
lugita15 said:Hmm, color me skeptical. Why can't you explicitly construct a sequence satisfying x_n \in (x - \frac{1}{n}, x + \frac{1}{n})? For instance, x_n can be the point a third of the way into the interval (x - \frac{1}{n}, x + \frac{1}{n}).