bounded sequence
A bounded sequence is one whose terms never wander outside a fixed corral. No matter how far you read into the sequence, every term stays between some floor and some ceiling. It does not have to settle down — it just is not allowed to escape to plus or minus infinity.
Formally, (a_n) is bounded if there is a real number M > 0 such that |a_n| <= M for all n. Equivalently, all terms lie in the interval [-M, M]. One can split this into bounded above (there is an upper bound: a_n <= U for all n) and bounded below (there is a lower bound: a_n >= L for all n); a sequence is bounded precisely when it is both.
Boundedness is the gatekeeper for two of the deepest results about sequences. Every convergent sequence is automatically bounded, which makes boundedness a quick necessary check for convergence. And by the Bolzano-Weierstrass theorem, every bounded real sequence — even a wildly oscillating one — has at least one convergent subsequence. But beware: boundedness alone does not give convergence, as (-1)^n shows.
a_n = sin(n) is bounded since |sin(n)| <= 1 for every n, so M = 1 works; yet it never converges. The bounded sequence a_n = (-1)^n + 1/n stays in [-1, 1.5] but also fails to converge.
Bounded is necessary for convergence but far from sufficient.
The bound M must work for all n at once. A sequence like a_n = n is unbounded even though each individual term is a finite number; no single M caps them all.