I've made a anki deck of Principles of Mathematical Analysis by Walter Rudin. The decks were generated by Gemini. If there are any suggestions, improvements, errors. Please contact [email protected]
| Front | For a sequence of real numbers \(\{s_n\}\), define its lower limit (limit inferior), \(\liminf_{n\to\infty} s_n\). |
| Back | Let E be the set of all subsequential limits of \(\{s_n\}\) in the extended real number system. The lower limit, denoted \(s_*\), is defined as:\(s_* = \inf E\). |
| Front | What is the intuition behind the change of variables formula \(\int_A^B f(\phi(y)) \,d(\alpha(\phi(y))) = \int_a^b f(x) \,d\alpha(x)\)? |
| Back | The mapping \(\phi\) transforms the interval of integration from \([A, B]\) to \([a, b]\). The sums \( \sum g(y_j) \Delta\beta_j = \sum f(\phi(y_j)) [\alpha(\phi(y_j)) - \alpha(\phi(y_{j-1}))]\) for a partition of \([A, B]\) correspond directly to the sums \(\sum f(x_i) \Delta\alpha_i\) for the corresponding partition of \([a, b]\), where \(x_i = \phi(y_i)\). Since the sums are equal, their limits (the integrals) must be equal. |
| Front | How is the number π defined in the context of these trigonometric functions? |
| Back | It is defined as twice the smallest positive number \(x_0\) for which C(x₀) = 0. The existence of such a number is proven by showing C(x) must cross the x-axis. |