Divergent Sequence may be Bounded

Theorem
While every Convergent Sequence is Bounded, it does not follow that every bounded sequence is convergent.

That is, there exist bounded sequences which are divergent.

Also see

 * Definition:Grandi's Series, $\displaystyle \sum_{n \mathop = 0}^\infty \paren {-1}^n$, is based upon the sequence $\sequence {x_n}$ defined in proofs 1 and 2.