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 11 - Extreme and Intermediate Value Theorem; Metric Spaces |
Using sequences and properties of sequences, we show both the extreme value theorem as well as the intermediate value theorem. We also define the notion of a metric space. A metric space is a space with a well-defined way of measuring distances between pairs of points. Many of the notions that we have introduced for the real numbers have counterparts for metric spaces and we discuss some of those including convergence and Cauchy sequence. We will also give a number of examples of metric spaces.
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