Interactive Real Analysis - part of MathCS.org

• Real Analysis
• 1. Sets and Relations
• 2. Infinity and Induction
• 3. Sequences of Numbers
• 4. Series of Numbers
• 5. Topology
• 5.1. Open and Closed Sets
• 5.2. Compact and Perfect Sets
• 5.3. Connected and Disconnected Sets
• 6. Limits, Continuity, and Differentiation
• 7. The Integral
• 8. Sequences of Functions
• 9. Historical Tidbits
• Java Tools

Examples 5.2.2(a):

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Is the interval [0,1] compact ? How about [0, 1) ?

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• The interval [0, 1] is compact. To see this, take any sequence of points in [0, 1]. Since the sequence must be bounded, we can extract a convergent subsequence by the Bolzano Weierstrass theorem. Using the theorem on accumulation and boundary points and noting that the set [0, 1] is closed, the limit of this subsequence must be contained in [0, 1]. Hence, the set is compact by definition.
• The interval [0, 1) is not compact. Consider the sequence { 1 - 1/n }. Then that sequence is contained in [0, 1), and converges to 1. Therefore, every subsequence of it must also converge to 1, which is not part of the original set. Therefore, the set can not be compact.