complete (original) (raw)

Examples:

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Cauchy sequence

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  The space ℚ of rational numbers is not complete: the sequence 3, 3.1, 3.14, 3.141, 3.1415, 3.14159, 3.141592⁢… consisting of finite decimals converging to π∈ℝ is a Cauchy sequence in ℚ that does not converge in ℚ.
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  The space ℝ of real numbers is complete, as it is the completion of ℚ with respect to the standard metric (other completions, such as the p-adic numbers, are also possible). More generally, the completion of any metric space is a complete metric space.
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  Every Banach space is complete. For example, the Lp–space of p-integrable functions is a complete metric space if p≥1.