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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.