Interactive Real Analysis

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 2.1. Countable Infinity
• 2.2. Uncountable Infinity
• 2.3. The Principle of Induction
• 2.4. The Real Number System
• 2.5. Pi and e are irrational
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

2.4. The Real Number System

Examples 2.4.3(b):

Consider the set of rational numbers {1, 1.4, 1.41, 1.414, 1.4142, ...} converging to the square root of 2. If all we knew were rational numbers, this set would have no supremum. If we allow real numbers, there is a unique supremem.

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• No number bigger than is the least upper bound (although each of these numbers is an upper bound), because if x was that least upper bound, then we can find a rational number between and x. That rational number would then be an upper bound smaller than x, which is a contradiction.
• No number less than is the least upper bound, because if x was that least upper bound, there is some element of the set between x and . But then x is not an upper bound, which is a contradiction.

If we consider this set as a subset of the real numbers, then the least upper bound of this set is .