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.
A sequence of real numbers is bounded above if there is a real upper bound 
with 
for every 
. It is bounded below if there
is a real lower bound 
with 
for every 
. A real sequence
is bounded precisely when it has both bounds.
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. By the monotone sequence theorem,
a bounded monotonic sequence of real
numbers is a convergent sequence.
