An unbounded sequence is a sequence 
of real numbers or complex
numbers that is not a bounded sequence. Thus,
for every positive real number 
, some term satisfies 
(Lebl, §2.1).
An unbounded sequence need not tend to infinity. For example, 
is unbounded both above and below, whereas 
tends to positive infinity.
In particular, unboundedness alone does not imply that the absolute
values of the terms tend to infinity. The sequence
with 
and 
for 
is a counterexample. Every convergent
sequence is bounded, so every unbounded sequence
is divergent.
