Heine-Borel Theorem in R^n: The Heine-Borel theorem in n-dimensional Euclidean space (R^n) states that a metric space is complete if and only if it is compa...
Heine-Borel Theorem in R^n:
The Heine-Borel theorem in n-dimensional Euclidean space (R^n) states that a metric space is complete if and only if it is compact. This means that any Cauchy sequence in the space converges to a unique point.
In other words, a metric space is complete if every non-empty set has a closed neighborhood. A set is closed if it contains all its limit points. A sequence converges to a point if its every limit point belongs to the sequence.
Proof:
If a metric space is complete, it is compact. This is because a compact space is complete, meaning every Cauchy sequence converges to a point. A sequence in a compact space converges to a point if and only if every subsequence of the sequence converges to the same point. Since a Cauchy sequence is a sequence of points, this means that it converges to a point.
Conversely, if a metric space is compact, it is complete. This is because a compact space is complete, meaning every non-empty set has a closed neighborhood. A closed neighborhood is a set that contains all its limit points. Since every limit point belongs to the set, and every sequence in the space converges to a point, it follows that every Cauchy sequence converges to a point.
Examples:
Any Euclidean space (R^n) is complete. This is because every Cauchy sequence in R^n converges to a unique point.
The space of continuous functions on a closed interval [0, 1] with the metric of uniform convergence is complete.
The space of bounded functions on [0, 1] with the metric of maximum distance is compact.
The space of real numbers with the metric of the absolute value is not complete