Bounded sequence

Definition

A bounded sequence is a sequence of numbers where all elements lie within a fixed finite interval. Mathematically, a sequence $\{a_n\}$ is bounded if there exists real numbers $M$ and $m$ such that $m \leq a_n \leq M$ for all $n$.

5 Must Know Facts For Your Next Test

1. A sequence can be bounded above, below, or both.
2. If a sequence is convergent, it must be bounded.
3. Not all bounded sequences are convergent.
4. The Bolzano-Weierstrass theorem states that every bounded sequence has at least one convergent subsequence.
5. To show that a sequence is bounded, you need to find specific bounds $M$ and $m$.

Review Questions

• What does it mean for a sequence to be bounded?
• How can you prove that a given sequence is bounded?
• Is every bounded sequence also convergent? Why or why not?

Related terms

Convergent Sequence: A sequence whose terms approach a specific value as the number of terms goes to infinity.

Subsequence: A new sequence derived by selecting terms from another sequence without changing their order.

Bolzano-Weierstrass Theorem: A theorem stating that every bounded sequence in $\mathbb{R}$ has at least one convergent subsequence.