Are all convergent sequences bounded and monotone? I know of the monotone convergence theorem, but does this mean that sequences converge only if they are bounded and monotone?

In general, you can choose some Cauchy sequence $\{a_n\}$ in $\mathbb R$ which is alternating, or "jumps around." Then $\{a_n\}$ is bounded and converges but is not monotone.
For example:
$\{a_n\} = \frac 1n$ for odd $n$, $0$ for even $n$
$\{b_n\} = \frac 1 {n^2} \sin (n) $ 
