Is every bounded sequence convergent? It's true, that every convergent sequence is bounded, but is every bounded sequence convergent?

Hint: $\quad$Consider the sequence $\{a_n\},\;a_n = (-1)^n\,$
It is bounded in $[-1, 1]\; ($ indeed, $a_n \in \{-1, 1\}\; \forall a_n\in \{a_n\}),\;$ but $\;\lim_{n\to \infty} (-1)^n\;$ does not exist.
Note: it is true that every bounded sequence contains a convergent subsequence, and furthermore, every monotonic sequence converges if and only if it is bounded. 

Added
See the entry on the Monotone Convergence Theorem for more information on the guaranteed convergence of bounded monotone sequences.
