A bounded sequence is a sequence 
of real numbers or complex
numbers for which there is a finite constant 
such that 
for every 
. Equivalently, its terms form a bounded
set.
Every convergent sequence of real numbers or complex numbers is bounded, but
the converse is false. For example, 
is bounded but is not a convergent
sequence. The Bolzano-Weierstrass
theorem guarantees that every bounded sequence of real
numbers has a subsequence that is a convergent
sequence. A monotonic sequence of real
numbers that is bounded is a convergent sequence.
