Consider a sequence of real numbers $\{a_n\}$ and these 3 definitions:
(1) $\{a_n\}$ is bounded: $\exists k>0$ s.t. $-k \leq a_n\leq k$ $\forall n \in \mathbb{N}$
(2) $a_n=O(1)$: $\exists M>0, \bar{n}\in \mathbb{N}$ s.t. $-M\leq a_n\leq M$ $\forall n \geq \bar{n}$
(3) $\{a_n\}$ is convergent ($\lim_{n\rightarrow \infty} a_n=b$, $|b|<\infty$): $\forall \epsilon>0$, $\exists \bar{n}_\epsilon \in \mathbb{N}$ s.t. $-\epsilon<a_n-b<\epsilon$ $\forall n \geq \bar{n}_\epsilon$
I think that
(a) (3) implies (1). In particular, if $\{a_n\}$ is monotone then $k=b$
(b) If $\{a_n\}$ is monotone then (1) implies (3)
(c) (1) implies (2)
(d) (2) does not imply (1)
(e) (3) implies (2)
(f) (2) does not imply (3)
Are these conclusions correct? Do you see other relations among the three concepts that I have missed?
(a) "In particular" is wrong: take $a_{n}=-1+1/n$ then $b=-1$ and (potentially) every $k\ge 1$ works.
(b)-(c) Correct.
(d) Wrong: (1) and (2) are equivalent.
(e)-(f) Correct.