Bounded below

Definition

A sequence is bounded below if there exists a real number that is less than or equal to every term in the sequence. This means that the terms of the sequence never fall below this specific value.

5 Must Know Facts For Your Next Test

1. A sequence \( \{a_n\} \) is bounded below if there exists a lower bound \( L \) such that \( a_n \geq L \) for all terms in the sequence.
2. The lower bound does not need to be unique; any number less than or equal to all terms of the sequence qualifies as a lower bound.
3. Being bounded below does not imply convergence, but it is a necessary condition for convergence in many contexts.
4. The concept of being bounded below can be applied to both finite and infinite sequences.
5. In mathematical notation, if a sequence is bounded below by \( L \), we write: $\exists L \,\text{such that}\, a_n \geq L \,\forall n$.

Review Questions

Related terms

Bounded Above: A sequence is bounded above if there exists a real number that is greater than or equal to every term in the sequence.

Convergence:

A sequence converges if its terms approach a specific value as they progress towards infinity.

Monotonic Sequence: A monotonic sequence is one that is either entirely non-increasing or non-decreasing.