The Bolzano-Weierstrass Theorem states that every closed and bounded subset of Euclidean space ℝⁿ is sequentially compact. This means that any infinite sequence of points within such a set has a subsequence that converges to a point also within the set.
Conditions: S is a subset of n-dimensional real space ℝⁿ.; S is closed.; S is bounded.
The Bolzano-Weierstrass Theorem states that every closed and bounded subset of Euclidean space ℝⁿ is sequentially compact. This means that any infinite sequence of points within such a set has a subsequence that converges to a point also within the set.
Conditions: S is a subset of n-dimensional real space ℝⁿ.; S is closed.; S is bounded.
The Heine-Borel Theorem states that in finite-dimensional Euclidean space ℝⁿ, a subset is compact if and only if it is both closed and bounded. This provides a practical criterion for identifying compact sets in standard geometric spaces.
Conditions: The space is finite-dimensional Euclidean space ℝⁿ.; S is a subset of ℝⁿ.
The Heine-Borel Theorem states that in finite-dimensional Euclidean space ℝⁿ, a subset is compact if and only if it is both closed and bounded. This provides a practical criterion for identifying compact sets in standard geometric spaces.
Conditions: The space is finite-dimensional Euclidean space ℝⁿ.; S is a subset of ℝⁿ.