bounded set

A bounded set is one that fits inside a finite window: there is a left wall and a right wall, and the whole set lives between them. Nothing in the set runs off to plus or minus infinity.

Precisely, a set S of reals is bounded if it is both bounded above and bounded below, that is, there are numbers m and M with m at most x at most M for every x in S. An equivalent single condition is that there is a number R with |x| at most R for all x in S, so the entire set sits inside the interval from minus R to R.

Boundedness is a recurring hypothesis throughout analysis. A nonempty bounded set of reals automatically has both a supremum and an infimum by completeness; a bounded monotone sequence converges; and by Bolzano–Weierstrass every bounded sequence has a convergent subsequence. Boundedness is often exactly the condition that prevents escape to infinity.

Be careful: bounded is not the same as finite, and not the same as closed. The interval from 0 to 1 is bounded but infinite, and the open interval from 0 to 1 is bounded but not closed. It is the combination bounded and closed that, on the line, gives compactness.