Certainly! Let's analyze the statement presented in the image step by step: --- ### **Step 1: Understanding the Lemma Setup** The lemma states: Let \( S \) be a subset of a metric space \((X, d)\). It is considering the property of "total boundedness" for the subset \( S \). **Explanation:** Here, \( (X, d) \) is a metric space, meaning it is a set \( X \) equipped with a distance function \( d \). The subset \( S \) is being considered with respect to whether it is "totally bounded" in the sense given by a previous definition (likely, that for every \( \epsilon > 0 \), \( S \) can be covered by finitely many balls of radius \( \epsilon \)). --- ### **Step 2: Stating the Condition and Its Equivalence** Condition (a) says: \( S \) is totally bounded, i.e., the metric space \((S, d_S)\) is totally bounded as per Definition 02.6. **Explanation:** This means that the subset \( S \), when viewed as a metric space with the induced metric \( d_S \) from \( X \), satisfies the property that for every \( \epsilon > 0 \), there exists a finite cover of \( S \) by open balls of radius \( \epsilon \). The lemma states that this condition is equivalent to some others (not shown in the image). --- ### **Final Answer:** The lemma is stating that for a subset \( S \) of a metric space \((X, d)\), the condition that \( S \) is totally bounded (i.e., can be covered by finitely many \(\epsilon\)-balls for any \(\epsilon > 0\)) is equivalent to other conditions (not shown in the image). The explanation after each step clarifies the meaning of the terms and what the lemma is asserting.