A **lower bound** of a set \( S \) of real numbers is a value \( m \) such that \( m \leq x \) for all elements \( x \) in the set \( S \). In other words, \( m \) is less than or equal to every element in \( S \). 

For example, if \( S = \{3, 5, 7\} \), then any number less than or equal to 3 is a lower bound of \( S \). Specifically, \( m = 3 \) is called the **greatest lower bound** or **infimum** of \( S \), since it is the largest value that is still a lower bound.

### Key Points:
- A lower bound does not have to be a member of the set.
- There can be multiple lower bounds for a set, but the greatest lower bound (infimum) is unique.
- If a set is bounded below, it has at least one lower bound.

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### Related Questions:
1. What is an upper bound of a set?
2. What is the difference between a lower bound and the infimum?
3. How do you determine if a set is bounded below?
4. Can a set have both a lower bound and an upper bound?
5. How does the concept of a lower bound apply in optimization problems?
6. What is the significance of the infimum in real analysis?
7. How do you calculate the lower bound of an interval?
8. Can a set have no lower bound?

#### Tip:
When analyzing bounds, it's useful to visualize the set on a number line to clearly see where the lower and upper bounds lie.

Explore the concept of lower bounds in set theory and real numbers. Learn what constitutes a lower bound, including the definition and examples. This article covers key points such as the greatest lower bound (infimum) and its significance in mathematics.

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