Definition:Bounded Above Sequence/Real

This page is about Bounded Above Real Sequence. For other uses, see Bounded Above.

Definition

Let $\sequence {x_n}$ be a real sequence.

Then $\sequence {x_n}$ is bounded above if and only if:

$\exists M \in \R: \forall i \in \N: x_i \le M$

Unbounded Above

$\sequence {x_n}$ is unbounded above if and only if there exists no $M$ in $\R$ such that:

$\forall i \in \N: x_i \le M$

Also see

• Definition:Bounded Below Real Sequence
• Definition:Bounded Real Sequence

• Definition:Bounded Above Real-Valued Function, of which a bounded above real sequence is the special case where the domain of the real-valued function is $\N$.

• Definition:Lower Bound of Real Sequence

Sources

• 1975: Bert Mendelson: Introduction to Topology (3rd ed.) ... (previous) ... (next): Chapter $2$: Metric Spaces: $\S 5$: Limits: Exercise $4$
• 1977: K.G. Binmore: Mathematical Analysis: A Straightforward Approach ... (previous) ... (next): $\S 4$: Convergent Sequences: $\S 4.2$: Sequences
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): bounded sequence
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): bounded sequence