18.100B Real Analysis
18.100B Real Analysis (Spring 2025, MIT OCW). Instructor: Prof. Tobias Holck Colding. This course gives an introduction to analysis, and the goal is twofold:
1. To learn how to prove mathematical theorems in analysis and how to write proofs.
2. To prove theorems in calculus in a rigorous way.
The course will start with real numbers, limits, convergence, series and continuity. We will continue on with metric spaces, differentiation and Riemann integrals. After that, we will move on to differential equations.
(from ocw.mit.edu)
| Lecture 07 - Bolzano-Weierstrass Theorem; Cauchy Sequences; Series |
The Bolzano-Weierstrass theorem says that any bounded sequence has a convergent subsequence. This crucial fact can be used to show other important theorems. An almost immediate consequence of it is the Cauchy convergence theorem. We also introduce the notion of a series and get acquainted with perhaps the single most important series: the geometric series. For a series, the most important question is whether or not it is convergent. To determine that, there are a number of tests beginning with the two versions of the comparison test.
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