Bounded Sequences

Definition

A sequence is considered bounded if its terms are restricted from growing infinitely large or infinitely small. Specifically, a sequence $a_n$ is bounded above if there is a real number $M$ such that $a_n \leq M$ for all $n$, and bounded below if there is a real number $m$ such that $a_n \geq m$ for all $n$. If it is bounded both above and below, it is simply called a bounded sequence.

$m \leq a_n \leq M \quad \text{for all } n$ How to read: m is less than or equal to a sub n, which is less than or equal to M for all n. Meaning / when to use: This inequality is used to mathematically prove that a sequence never exceeds a specific upper limit $M$ or drops below a specific lower limit $m$.

Why It Matters

Boundedness is a fundamental guarantee of stability. In computational algorithms, proving that a sequence of error terms or outputs is bounded ensures that the program will not crash due to numerical overflow. In theoretical mathematics, boundedness is a prerequisite for convergence; an unbounded sequence cannot converge to a finite limit. Without bounded structures, systems escalate out of control.

Core Concepts

• Upper Bound: A ceiling value that the sequence never exceeds. The smallest possible ceiling is the supremum.
• Lower Bound: A floor value that the sequence never drops below. The largest possible floor is the infimum.
• Monotone Convergence Theorem: If a sequence is both bounded and monotonic (either entirely non-increasing or non-decreasing), it must converge to a limit.
• System Constraints: In applied modeling, physical limits (like the speed of light or absolute zero) act as natural bounds on sequences of system states.

Connected Concepts

• Convergence
• Divergence
• Monotonicity
• Upper Bound
• Limit Laws Sequences