This question asks to identify a key theorem from real analysis by its statement. This theorem is fundamental to the concept of compactness.
Step 1: Understanding the Question:
We need to name the theorem which guarantees that any sequence confined within a finite region (bounded) of \( \mathbb{R}^n \) must have a subsequence that "settles down" to a specific point (converges).
Step 2: Key Formula or Approach:
This requires recognizing the statement of the Bolzano-Weierstrass Theorem and distinguishing it from other major theorems in calculus.
Step 3: Detailed Explanation:
Let's analyze the theorems listed:
Mean Value Theorem: Relates the average rate of change of a function over an interval to its instantaneous rate of change (derivative) at some point in that interval.
Bolzano–Weierstrass Theorem: This theorem states exactly what is described in the question: every bounded sequence in \( \mathbb{R}^n \) has a convergent subsequence. It essentially says that if you have an infinite number of points in a finite space, they must "cluster" around at least one point.
Rolle’s Theorem: A special case of the Mean Value Theorem, stating that if a differentiable function has equal values at two points, there must be a point between them where the derivative is zero.
Taylor’s Theorem: Provides a way to approximate a function near a point using a polynomial whose coefficients are derived from the function's derivatives at that point.
The statement in the question directly matches the definition of the Bolzano-Weierstrass Theorem.
Step 4: Final Answer:
The theorem is the Bolzano–Weierstrass Theorem.